Properties

Label 8-570e4-1.1-c7e4-0-0
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $1.00521\times 10^{9}$
Root an. cond. $13.3438$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·2-s − 108·3-s + 640·4-s + 500·5-s − 3.45e3·6-s − 1.49e3·7-s + 1.02e4·8-s + 7.29e3·9-s + 1.60e4·10-s − 2.91e3·11-s − 6.91e4·12-s − 3.69e3·13-s − 4.78e4·14-s − 5.40e4·15-s + 1.43e5·16-s + 16·17-s + 2.33e5·18-s + 2.74e4·19-s + 3.20e5·20-s + 1.61e5·21-s − 9.31e4·22-s − 7.30e4·23-s − 1.10e6·24-s + 1.56e5·25-s − 1.18e5·26-s − 3.93e5·27-s − 9.57e5·28-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s − 1.64·7-s + 7.07·8-s + 10/3·9-s + 5.05·10-s − 0.659·11-s − 11.5·12-s − 0.466·13-s − 4.66·14-s − 4.13·15-s + 35/4·16-s + 0.000789·17-s + 9.42·18-s + 0.917·19-s + 8.94·20-s + 3.80·21-s − 1.86·22-s − 1.25·23-s − 16.3·24-s + 2·25-s − 1.31·26-s − 3.84·27-s − 8.24·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.00521\times 10^{9}\)
Root analytic conductor: \(13.3438\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

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$C_1$ \( ( 1 - p^{3} T )^{4} \)
3$C_1$ \( ( 1 + p^{3} T )^{4} \)
5$C_1$ \( ( 1 - p^{3} T )^{4} \)
19$C_1$ \( ( 1 - p^{3} T )^{4} \)
good7$C_2 \wr S_4$ \( 1 + 1496 T + 1348 p^{4} T^{2} + 3248754112 T^{3} + 4054349586902 T^{4} + 3248754112 p^{7} T^{5} + 1348 p^{18} T^{6} + 1496 p^{21} T^{7} + p^{28} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 2912 T + 6299692 p T^{2} + 170974862232 T^{3} + 176507609995122 p T^{4} + 170974862232 p^{7} T^{5} + 6299692 p^{15} T^{6} + 2912 p^{21} T^{7} + p^{28} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 3696 T + 184023652 T^{2} + 254871226936 T^{3} + 14516361784261494 T^{4} + 254871226936 p^{7} T^{5} + 184023652 p^{14} T^{6} + 3696 p^{21} T^{7} + p^{28} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 16 T + 857017292 T^{2} + 11057072782992 T^{3} + 325557069946951974 T^{4} + 11057072782992 p^{7} T^{5} + 857017292 p^{14} T^{6} - 16 p^{21} T^{7} + p^{28} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 73016 T + 10463536220 T^{2} + 652431450973848 T^{3} + 49932867735426482982 T^{4} + 652431450973848 p^{7} T^{5} + 10463536220 p^{14} T^{6} + 73016 p^{21} T^{7} + p^{28} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 137784 T + 42386981396 T^{2} + 2375840362140720 T^{3} + \)\(71\!\cdots\!34\)\( T^{4} + 2375840362140720 p^{7} T^{5} + 42386981396 p^{14} T^{6} + 137784 p^{21} T^{7} + p^{28} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 198072 T + 85820070508 T^{2} - 12336940811511128 T^{3} + \)\(33\!\cdots\!42\)\( T^{4} - 12336940811511128 p^{7} T^{5} + 85820070508 p^{14} T^{6} - 198072 p^{21} T^{7} + p^{28} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 207256 T + 152302694116 T^{2} - 26945217679675184 T^{3} + \)\(10\!\cdots\!22\)\( T^{4} - 26945217679675184 p^{7} T^{5} + 152302694116 p^{14} T^{6} - 207256 p^{21} T^{7} + p^{28} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 504056 T + 658336935524 T^{2} + 247661304371015808 T^{3} + \)\(17\!\cdots\!66\)\( T^{4} + 247661304371015808 p^{7} T^{5} + 658336935524 p^{14} T^{6} + 504056 p^{21} T^{7} + p^{28} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 250368 T + 794702741284 T^{2} + 114233542183566952 T^{3} + \)\(28\!\cdots\!38\)\( T^{4} + 114233542183566952 p^{7} T^{5} + 794702741284 p^{14} T^{6} + 250368 p^{21} T^{7} + p^{28} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 1000376 T + 2324322260924 T^{2} + 1544686466658023256 T^{3} + \)\(18\!\cdots\!06\)\( T^{4} + 1544686466658023256 p^{7} T^{5} + 2324322260924 p^{14} T^{6} + 1000376 p^{21} T^{7} + p^{28} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 2178688 T + 5028794631452 T^{2} + 7076462590603421760 T^{3} + \)\(90\!\cdots\!66\)\( T^{4} + 7076462590603421760 p^{7} T^{5} + 5028794631452 p^{14} T^{6} + 2178688 p^{21} T^{7} + p^{28} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 327976 T + 6413641153772 T^{2} + 1443811406273933784 T^{3} + \)\(18\!\cdots\!22\)\( T^{4} + 1443811406273933784 p^{7} T^{5} + 6413641153772 p^{14} T^{6} - 327976 p^{21} T^{7} + p^{28} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 572936 T + 1054077021484 T^{2} + 2799262298341413160 T^{3} - \)\(40\!\cdots\!98\)\( T^{4} + 2799262298341413160 p^{7} T^{5} + 1054077021484 p^{14} T^{6} - 572936 p^{21} T^{7} + p^{28} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 2017152 T + 20671485487084 T^{2} - 28219577873060887232 T^{3} + \)\(17\!\cdots\!06\)\( T^{4} - 28219577873060887232 p^{7} T^{5} + 20671485487084 p^{14} T^{6} - 2017152 p^{21} T^{7} + p^{28} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 2828960 T + 15546196545692 T^{2} - 59319424112529703200 T^{3} + \)\(19\!\cdots\!82\)\( T^{4} - 59319424112529703200 p^{7} T^{5} + 15546196545692 p^{14} T^{6} - 2828960 p^{21} T^{7} + p^{28} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 132392 T + 27176541756604 T^{2} - 7773597997820623208 T^{3} + \)\(36\!\cdots\!02\)\( T^{4} - 7773597997820623208 p^{7} T^{5} + 27176541756604 p^{14} T^{6} + 132392 p^{21} T^{7} + p^{28} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 3418408 T + 22454879188540 T^{2} - \)\(13\!\cdots\!52\)\( T^{3} + \)\(52\!\cdots\!90\)\( T^{4} - \)\(13\!\cdots\!52\)\( p^{7} T^{5} + 22454879188540 p^{14} T^{6} - 3418408 p^{21} T^{7} + p^{28} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 3201760 T + 54189243883340 T^{2} + \)\(25\!\cdots\!52\)\( T^{3} + \)\(16\!\cdots\!42\)\( T^{4} + \)\(25\!\cdots\!52\)\( p^{7} T^{5} + 54189243883340 p^{14} T^{6} + 3201760 p^{21} T^{7} + p^{28} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1389392 T + 147421694160644 T^{2} + \)\(15\!\cdots\!32\)\( T^{3} + \)\(92\!\cdots\!10\)\( T^{4} + \)\(15\!\cdots\!32\)\( p^{7} T^{5} + 147421694160644 p^{14} T^{6} + 1389392 p^{21} T^{7} + p^{28} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 21061144 T + 447143586952804 T^{2} + \)\(52\!\cdots\!04\)\( T^{3} + \)\(58\!\cdots\!06\)\( T^{4} + \)\(52\!\cdots\!04\)\( p^{7} T^{5} + 447143586952804 p^{14} T^{6} + 21061144 p^{21} T^{7} + p^{28} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.83542422842482173020552360828, −6.46226604641903186924078677140, −6.38562358613559553808510551782, −6.24672068484094918018102286802, −6.17593381554924944032545108062, −5.67418093443114627156839170036, −5.50448558614606393580776015377, −5.50362967236820008764742892772, −5.28568884354088420674156660131, −4.89623264499621618867579917702, −4.66802448040047593795174210712, −4.66126442781436518906935226775, −4.46572667909353091368806158119, −3.64567164590210469820159543188, −3.64131959955026179367374751497, −3.49335427051432063136586913920, −3.43757940186406572271588510180, −2.61672442849577308560504444531, −2.46792239455251007111077624531, −2.44859865232308545145693404211, −2.32759978187260714475029258538, −1.40980691631626186529099139750, −1.34399828320283444445759141762, −1.29304625500940365088491598912, −1.22754601498258425998302544240, 0, 0, 0, 0, 1.22754601498258425998302544240, 1.29304625500940365088491598912, 1.34399828320283444445759141762, 1.40980691631626186529099139750, 2.32759978187260714475029258538, 2.44859865232308545145693404211, 2.46792239455251007111077624531, 2.61672442849577308560504444531, 3.43757940186406572271588510180, 3.49335427051432063136586913920, 3.64131959955026179367374751497, 3.64567164590210469820159543188, 4.46572667909353091368806158119, 4.66126442781436518906935226775, 4.66802448040047593795174210712, 4.89623264499621618867579917702, 5.28568884354088420674156660131, 5.50362967236820008764742892772, 5.50448558614606393580776015377, 5.67418093443114627156839170036, 6.17593381554924944032545108062, 6.24672068484094918018102286802, 6.38562358613559553808510551782, 6.46226604641903186924078677140, 6.83542422842482173020552360828

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