Properties

Label 10-76e5-1.1-c7e5-0-0
Degree $10$
Conductor $2535525376$
Sign $-1$
Analytic cond. $7.54256\times 10^{6}$
Root an. cond. $4.87250$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·3-s − 280·5-s + 414·7-s − 3.48e3·9-s − 2.66e3·11-s − 602·13-s + 3.92e3·15-s − 2.73e4·17-s − 3.42e4·19-s − 5.79e3·21-s − 6.70e4·23-s − 2.10e5·25-s + 248·27-s − 3.72e5·29-s − 2.71e5·31-s + 3.72e4·33-s − 1.15e5·35-s − 5.62e5·37-s + 8.42e3·39-s − 9.56e5·41-s − 8.27e5·43-s + 9.74e5·45-s − 1.81e6·47-s − 2.40e6·49-s + 3.83e5·51-s + 4.86e5·53-s + 7.45e5·55-s + ⋯
L(s)  = 1  − 0.299·3-s − 1.00·5-s + 0.456·7-s − 1.59·9-s − 0.603·11-s − 0.0759·13-s + 0.299·15-s − 1.35·17-s − 1.14·19-s − 0.136·21-s − 1.14·23-s − 2.69·25-s + 0.00242·27-s − 2.83·29-s − 1.63·31-s + 0.180·33-s − 0.457·35-s − 1.82·37-s + 0.0227·39-s − 2.16·41-s − 1.58·43-s + 1.59·45-s − 2.54·47-s − 2.91·49-s + 0.404·51-s + 0.449·53-s + 0.604·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 19^{5}\)
Sign: $-1$
Analytic conductor: \(7.54256\times 10^{6}\)
Root analytic conductor: \(4.87250\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{76} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{10} \cdot 19^{5} ,\ ( \ : 7/2, 7/2, 7/2, 7/2, 7/2 ),\ -1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
19$C_1$ \( ( 1 + p^{3} T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + 14 T + 3676 T^{2} + 11104 p^{2} T^{3} + 247145 p^{3} T^{4} + 9484444 p^{3} T^{5} + 247145 p^{10} T^{6} + 11104 p^{16} T^{7} + 3676 p^{21} T^{8} + 14 p^{28} T^{9} + p^{35} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 56 p T + 57814 p T^{2} + 2652558 p^{2} T^{3} + 1563182069 p^{2} T^{4} + 55600139212 p^{3} T^{5} + 1563182069 p^{9} T^{6} + 2652558 p^{16} T^{7} + 57814 p^{22} T^{8} + 56 p^{29} T^{9} + p^{35} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 414 T + 2575571 T^{2} - 2009518362 T^{3} + 2876468268073 T^{4} - 2752968688370496 T^{5} + 2876468268073 p^{7} T^{6} - 2009518362 p^{14} T^{7} + 2575571 p^{21} T^{8} - 414 p^{28} T^{9} + p^{35} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 2 p^{3} T + 47532404 T^{2} + 196653658836 T^{3} + 1564587459786083 T^{4} + 4588703502231464828 T^{5} + 1564587459786083 p^{7} T^{6} + 196653658836 p^{14} T^{7} + 47532404 p^{21} T^{8} + 2 p^{31} T^{9} + p^{35} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 602 T + 147508536 T^{2} + 786462907278 T^{3} + 9885416966268075 T^{4} + 88572120386284541784 T^{5} + 9885416966268075 p^{7} T^{6} + 786462907278 p^{14} T^{7} + 147508536 p^{21} T^{8} + 602 p^{28} T^{9} + p^{35} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 27366 T + 1416771931 T^{2} + 30995036043348 T^{3} + 976315396052141173 T^{4} + \)\(16\!\cdots\!54\)\( T^{5} + 976315396052141173 p^{7} T^{6} + 30995036043348 p^{14} T^{7} + 1416771931 p^{21} T^{8} + 27366 p^{28} T^{9} + p^{35} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 67096 T + 672690382 p T^{2} + 742746864613344 T^{3} + 97403622365062360565 T^{4} + \)\(15\!\cdots\!64\)\( p T^{5} + 97403622365062360565 p^{7} T^{6} + 742746864613344 p^{14} T^{7} + 672690382 p^{22} T^{8} + 67096 p^{28} T^{9} + p^{35} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 372398 T + 123594633620 T^{2} + 24322905221765418 T^{3} + \)\(45\!\cdots\!35\)\( T^{4} + \)\(60\!\cdots\!48\)\( T^{5} + \)\(45\!\cdots\!35\)\( p^{7} T^{6} + 24322905221765418 p^{14} T^{7} + 123594633620 p^{21} T^{8} + 372398 p^{28} T^{9} + p^{35} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 271372 T + 112879086507 T^{2} + 19060065197093520 T^{3} + \)\(49\!\cdots\!98\)\( T^{4} + \)\(64\!\cdots\!92\)\( T^{5} + \)\(49\!\cdots\!98\)\( p^{7} T^{6} + 19060065197093520 p^{14} T^{7} + 112879086507 p^{21} T^{8} + 271372 p^{28} T^{9} + p^{35} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 562630 T + 503027979609 T^{2} + 5044147202230440 p T^{3} + \)\(94\!\cdots\!90\)\( T^{4} + \)\(25\!\cdots\!60\)\( T^{5} + \)\(94\!\cdots\!90\)\( p^{7} T^{6} + 5044147202230440 p^{15} T^{7} + 503027979609 p^{21} T^{8} + 562630 p^{28} T^{9} + p^{35} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 956714 T + 858829677401 T^{2} + 558274454732676000 T^{3} + \)\(30\!\cdots\!54\)\( T^{4} + \)\(14\!\cdots\!52\)\( T^{5} + \)\(30\!\cdots\!54\)\( p^{7} T^{6} + 558274454732676000 p^{14} T^{7} + 858829677401 p^{21} T^{8} + 956714 p^{28} T^{9} + p^{35} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 827362 T + 1446200884344 T^{2} + 842028213283761024 T^{3} + \)\(80\!\cdots\!43\)\( T^{4} + \)\(33\!\cdots\!88\)\( T^{5} + \)\(80\!\cdots\!43\)\( p^{7} T^{6} + 842028213283761024 p^{14} T^{7} + 1446200884344 p^{21} T^{8} + 827362 p^{28} T^{9} + p^{35} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 1812982 T + 2671332480764 T^{2} + 2520094723128764352 T^{3} + \)\(23\!\cdots\!95\)\( T^{4} + \)\(16\!\cdots\!84\)\( T^{5} + \)\(23\!\cdots\!95\)\( p^{7} T^{6} + 2520094723128764352 p^{14} T^{7} + 2671332480764 p^{21} T^{8} + 1812982 p^{28} T^{9} + p^{35} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 486998 T + 4629026089784 T^{2} - 1938463029671914986 T^{3} + \)\(98\!\cdots\!83\)\( T^{4} - \)\(32\!\cdots\!92\)\( T^{5} + \)\(98\!\cdots\!83\)\( p^{7} T^{6} - 1938463029671914986 p^{14} T^{7} + 4629026089784 p^{21} T^{8} - 486998 p^{28} T^{9} + p^{35} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 367182 T - 1643557954748 T^{2} - 6627210193123102656 T^{3} + \)\(41\!\cdots\!31\)\( T^{4} + \)\(14\!\cdots\!44\)\( T^{5} + \)\(41\!\cdots\!31\)\( p^{7} T^{6} - 6627210193123102656 p^{14} T^{7} - 1643557954748 p^{21} T^{8} + 367182 p^{28} T^{9} + p^{35} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 1879732 T + 7749404395290 T^{2} - 9963704865691638198 T^{3} + \)\(38\!\cdots\!65\)\( T^{4} - \)\(47\!\cdots\!92\)\( T^{5} + \)\(38\!\cdots\!65\)\( p^{7} T^{6} - 9963704865691638198 p^{14} T^{7} + 7749404395290 p^{21} T^{8} - 1879732 p^{28} T^{9} + p^{35} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 1046394 T + 10097473571576 T^{2} + 25105239457925557932 T^{3} + \)\(64\!\cdots\!63\)\( T^{4} + \)\(25\!\cdots\!36\)\( T^{5} + \)\(64\!\cdots\!63\)\( p^{7} T^{6} + 25105239457925557932 p^{14} T^{7} + 10097473571576 p^{21} T^{8} + 1046394 p^{28} T^{9} + p^{35} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 4664572 T + 12023820502163 T^{2} + 22827464116687888320 T^{3} + \)\(99\!\cdots\!02\)\( T^{4} + \)\(47\!\cdots\!84\)\( T^{5} + \)\(99\!\cdots\!02\)\( p^{7} T^{6} + 22827464116687888320 p^{14} T^{7} + 12023820502163 p^{21} T^{8} + 4664572 p^{28} T^{9} + p^{35} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 4224942 T + 41896684548731 T^{2} - \)\(13\!\cdots\!28\)\( T^{3} + \)\(81\!\cdots\!65\)\( T^{4} - \)\(20\!\cdots\!14\)\( T^{5} + \)\(81\!\cdots\!65\)\( p^{7} T^{6} - \)\(13\!\cdots\!28\)\( p^{14} T^{7} + 41896684548731 p^{21} T^{8} - 4224942 p^{28} T^{9} + p^{35} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 9574024 T + 1156078839417 p T^{2} - \)\(53\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!38\)\( T^{4} - \)\(13\!\cdots\!04\)\( T^{5} + \)\(29\!\cdots\!38\)\( p^{7} T^{6} - \)\(53\!\cdots\!00\)\( p^{14} T^{7} + 1156078839417 p^{22} T^{8} - 9574024 p^{28} T^{9} + p^{35} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 11754804 T + 110268615554371 T^{2} - \)\(80\!\cdots\!76\)\( T^{3} + \)\(51\!\cdots\!90\)\( T^{4} - \)\(27\!\cdots\!08\)\( T^{5} + \)\(51\!\cdots\!90\)\( p^{7} T^{6} - \)\(80\!\cdots\!76\)\( p^{14} T^{7} + 110268615554371 p^{21} T^{8} - 11754804 p^{28} T^{9} + p^{35} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 2782542 T + 110940525514693 T^{2} - \)\(23\!\cdots\!64\)\( T^{3} + \)\(78\!\cdots\!14\)\( T^{4} - \)\(16\!\cdots\!36\)\( T^{5} + \)\(78\!\cdots\!14\)\( p^{7} T^{6} - \)\(23\!\cdots\!64\)\( p^{14} T^{7} + 110940525514693 p^{21} T^{8} - 2782542 p^{28} T^{9} + p^{35} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 1291574 T + 106423285884609 T^{2} + \)\(22\!\cdots\!16\)\( T^{3} + \)\(91\!\cdots\!50\)\( T^{4} + \)\(61\!\cdots\!12\)\( T^{5} + \)\(91\!\cdots\!50\)\( p^{7} T^{6} + \)\(22\!\cdots\!16\)\( p^{14} T^{7} + 106423285884609 p^{21} T^{8} - 1291574 p^{28} T^{9} + p^{35} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.013659569154647037090963357717, −7.998995673142444779296141374813, −7.982076131365813559284209642941, −7.939396058304497815093906973133, −7.49401862123509718419737984826, −7.14087023433922310854911057027, −6.67284926102135788254185532033, −6.50526427399407943655478232706, −6.28562746600711123647841210183, −6.21747090570803598503663720066, −5.50022837011900447565352460731, −5.48429504003763877122652506159, −5.11573175392536565095971963215, −5.05865578111671347205654370860, −4.72945928646603151656852285482, −4.00684480272684649751168576585, −3.98567431597430609171657104326, −3.49246247150725732406423722658, −3.42736479409068320434349872190, −3.40615805873004771621698979568, −2.36430134544470230845786203419, −2.10847702965545947670379304022, −2.08106368941208712746800657076, −1.64709687339730579241851273778, −1.43389284203445057390027421117, 0, 0, 0, 0, 0, 1.43389284203445057390027421117, 1.64709687339730579241851273778, 2.08106368941208712746800657076, 2.10847702965545947670379304022, 2.36430134544470230845786203419, 3.40615805873004771621698979568, 3.42736479409068320434349872190, 3.49246247150725732406423722658, 3.98567431597430609171657104326, 4.00684480272684649751168576585, 4.72945928646603151656852285482, 5.05865578111671347205654370860, 5.11573175392536565095971963215, 5.48429504003763877122652506159, 5.50022837011900447565352460731, 6.21747090570803598503663720066, 6.28562746600711123647841210183, 6.50526427399407943655478232706, 6.67284926102135788254185532033, 7.14087023433922310854911057027, 7.49401862123509718419737984826, 7.939396058304497815093906973133, 7.982076131365813559284209642941, 7.998995673142444779296141374813, 8.013659569154647037090963357717

Graph of the $Z$-function along the critical line