Properties

Label 8-210e4-1.1-c11e4-0-0
Degree $8$
Conductor $1944810000$
Sign $1$
Analytic cond. $6.77794\times 10^{8}$
Root an. cond. $12.7024$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 128·2-s + 972·3-s + 1.02e4·4-s + 1.25e4·5-s + 1.24e5·6-s + 6.72e4·7-s + 6.55e5·8-s + 5.90e5·9-s + 1.60e6·10-s + 4.58e5·11-s + 9.95e6·12-s + 1.57e6·13-s + 8.60e6·14-s + 1.21e7·15-s + 3.67e7·16-s + 8.67e6·17-s + 7.55e7·18-s + 1.24e7·19-s + 1.28e8·20-s + 6.53e7·21-s + 5.86e7·22-s + 5.13e5·23-s + 6.37e8·24-s + 9.76e7·25-s + 2.01e8·26-s + 2.86e8·27-s + 6.88e8·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s + 1.51·7-s + 7.07·8-s + 10/3·9-s + 5.05·10-s + 0.857·11-s + 11.5·12-s + 1.17·13-s + 4.27·14-s + 4.13·15-s + 35/4·16-s + 1.48·17-s + 9.42·18-s + 1.15·19-s + 8.94·20-s + 3.49·21-s + 2.42·22-s + 0.0166·23-s + 16.3·24-s + 2·25-s + 3.32·26-s + 3.84·27-s + 7.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(6.77794\times 10^{8}\)
Root analytic conductor: \(12.7024\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\)

Particular Values

\(L(6)\) \(\approx\) \(1474.690904\)
\(L(\frac12)\) \(\approx\) \(1474.690904\)
\(L(\frac{13}{2})\) 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$C_1$ \( ( 1 - p^{5} T )^{4} \)
3$C_1$ \( ( 1 - p^{5} T )^{4} \)
5$C_1$ \( ( 1 - p^{5} T )^{4} \)
7$C_1$ \( ( 1 - p^{5} T )^{4} \)
good11$C_2 \wr S_4$ \( 1 - 41660 p T + 555427682412 T^{2} - 6716091718596444 p T^{3} + \)\(13\!\cdots\!22\)\( T^{4} - 6716091718596444 p^{12} T^{5} + 555427682412 p^{22} T^{6} - 41660 p^{34} T^{7} + p^{44} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 1574316 T + 4698724538756 T^{2} - 4144806283403141028 T^{3} + \)\(95\!\cdots\!90\)\( T^{4} - 4144806283403141028 p^{11} T^{5} + 4698724538756 p^{22} T^{6} - 1574316 p^{33} T^{7} + p^{44} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 8678072 T + 108830083516764 T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!02\)\( T^{4} - \)\(75\!\cdots\!20\)\( p^{11} T^{5} + 108830083516764 p^{22} T^{6} - 8678072 p^{33} T^{7} + p^{44} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 12442004 T + 463934845572076 T^{2} - \)\(42\!\cdots\!28\)\( T^{3} + \)\(80\!\cdots\!66\)\( T^{4} - \)\(42\!\cdots\!28\)\( p^{11} T^{5} + 463934845572076 p^{22} T^{6} - 12442004 p^{33} T^{7} + p^{44} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 513088 T + 230211228849468 T^{2} - \)\(24\!\cdots\!04\)\( T^{3} - \)\(48\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!04\)\( p^{11} T^{5} + 230211228849468 p^{22} T^{6} - 513088 p^{33} T^{7} + p^{44} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 58476696 T + 25009583021169356 T^{2} - \)\(15\!\cdots\!52\)\( T^{3} + \)\(32\!\cdots\!66\)\( T^{4} - \)\(15\!\cdots\!52\)\( p^{11} T^{5} + 25009583021169356 p^{22} T^{6} - 58476696 p^{33} T^{7} + p^{44} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 145189572 T + 64640957819813612 T^{2} - \)\(96\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!70\)\( T^{4} - \)\(96\!\cdots\!16\)\( p^{11} T^{5} + 64640957819813612 p^{22} T^{6} - 145189572 p^{33} T^{7} + p^{44} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 340912752 T + 449679410845919132 T^{2} - \)\(13\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!98\)\( T^{4} - \)\(13\!\cdots\!36\)\( p^{11} T^{5} + 449679410845919132 p^{22} T^{6} - 340912752 p^{33} T^{7} + p^{44} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 915147368 T + 199627096716234492 T^{2} + \)\(12\!\cdots\!28\)\( T^{3} - \)\(12\!\cdots\!86\)\( T^{4} + \)\(12\!\cdots\!28\)\( p^{11} T^{5} + 199627096716234492 p^{22} T^{6} - 915147368 p^{33} T^{7} + p^{44} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 462244024 T - 337499855021694484 T^{2} + \)\(58\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!82\)\( T^{4} + \)\(58\!\cdots\!80\)\( p^{11} T^{5} - 337499855021694484 p^{22} T^{6} - 462244024 p^{33} T^{7} + p^{44} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 19185320 p T + 3945897582754584156 T^{2} - \)\(70\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} - \)\(70\!\cdots\!32\)\( p^{11} T^{5} + 3945897582754584156 p^{22} T^{6} - 19185320 p^{34} T^{7} + p^{44} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 157945788 T + 15178943563572621524 T^{2} - \)\(14\!\cdots\!24\)\( T^{3} + \)\(15\!\cdots\!38\)\( T^{4} - \)\(14\!\cdots\!24\)\( p^{11} T^{5} + 15178943563572621524 p^{22} T^{6} + 157945788 p^{33} T^{7} + p^{44} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 2706989128 T + \)\(10\!\cdots\!48\)\( T^{2} - \)\(21\!\cdots\!96\)\( T^{3} + \)\(46\!\cdots\!02\)\( T^{4} - \)\(21\!\cdots\!96\)\( p^{11} T^{5} + \)\(10\!\cdots\!48\)\( p^{22} T^{6} - 2706989128 p^{33} T^{7} + p^{44} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 8740846920 T + 63563623150159198700 T^{2} - \)\(31\!\cdots\!76\)\( T^{3} + \)\(30\!\cdots\!10\)\( T^{4} - \)\(31\!\cdots\!76\)\( p^{11} T^{5} + 63563623150159198700 p^{22} T^{6} - 8740846920 p^{33} T^{7} + p^{44} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 5883134368 T + \)\(49\!\cdots\!60\)\( T^{2} - \)\(21\!\cdots\!88\)\( T^{3} + \)\(91\!\cdots\!02\)\( T^{4} - \)\(21\!\cdots\!88\)\( p^{11} T^{5} + \)\(49\!\cdots\!60\)\( p^{22} T^{6} - 5883134368 p^{33} T^{7} + p^{44} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 344015372 T + \)\(23\!\cdots\!40\)\( T^{2} - \)\(52\!\cdots\!68\)\( T^{3} + \)\(84\!\cdots\!98\)\( T^{4} - \)\(52\!\cdots\!68\)\( p^{11} T^{5} + \)\(23\!\cdots\!40\)\( p^{22} T^{6} - 344015372 p^{33} T^{7} + p^{44} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 10549706244 T + \)\(13\!\cdots\!68\)\( T^{2} - \)\(68\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!98\)\( T^{4} - \)\(68\!\cdots\!08\)\( p^{11} T^{5} + \)\(13\!\cdots\!68\)\( p^{22} T^{6} - 10549706244 p^{33} T^{7} + p^{44} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 430177976 T + \)\(14\!\cdots\!24\)\( T^{2} - \)\(32\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} - \)\(32\!\cdots\!68\)\( p^{11} T^{5} + \)\(14\!\cdots\!24\)\( p^{22} T^{6} + 430177976 p^{33} T^{7} + p^{44} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 28504941432 T + \)\(35\!\cdots\!76\)\( T^{2} - \)\(80\!\cdots\!12\)\( T^{3} + \)\(57\!\cdots\!74\)\( T^{4} - \)\(80\!\cdots\!12\)\( p^{11} T^{5} + \)\(35\!\cdots\!76\)\( p^{22} T^{6} - 28504941432 p^{33} T^{7} + p^{44} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 26763786680 T + \)\(79\!\cdots\!60\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(29\!\cdots\!38\)\( T^{4} - \)\(23\!\cdots\!40\)\( p^{11} T^{5} + \)\(79\!\cdots\!60\)\( p^{22} T^{6} - 26763786680 p^{33} T^{7} + p^{44} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 62389990476 T + \)\(25\!\cdots\!84\)\( T^{2} - \)\(13\!\cdots\!04\)\( T^{3} + \)\(25\!\cdots\!34\)\( T^{4} - \)\(13\!\cdots\!04\)\( p^{11} T^{5} + \)\(25\!\cdots\!84\)\( p^{22} T^{6} - 62389990476 p^{33} T^{7} + p^{44} T^{8} \)
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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

−7.19303835531287636026552760439, −6.36133432961955851676320359289, −6.36106368773154639889763098411, −6.22044722763440522778596529484, −6.11259066250300813454781854460, −5.31422801123583529060166105933, −5.30568889524918928707048148403, −5.21647879976231365614463724856, −5.06734687326506723533183134516, −4.32586117838845141454775603649, −4.15540382998205030557410637441, −4.08003171818926122835866094054, −4.01101456240679535039804698886, −3.24520096290250700960999465739, −3.19304481626714687008779985677, −3.00335825802653935839394822545, −2.89964946952927745519305271898, −2.20734595579419197270498067775, −2.14761690749961516454598485638, −1.93053451866200685530677773768, −1.89770109376302234754972884116, −1.10630427223516749449499218581, −1.08519008667689081681413372157, −0.939201699977615044625180610938, −0.925571833160381609004041824964, 0.925571833160381609004041824964, 0.939201699977615044625180610938, 1.08519008667689081681413372157, 1.10630427223516749449499218581, 1.89770109376302234754972884116, 1.93053451866200685530677773768, 2.14761690749961516454598485638, 2.20734595579419197270498067775, 2.89964946952927745519305271898, 3.00335825802653935839394822545, 3.19304481626714687008779985677, 3.24520096290250700960999465739, 4.01101456240679535039804698886, 4.08003171818926122835866094054, 4.15540382998205030557410637441, 4.32586117838845141454775603649, 5.06734687326506723533183134516, 5.21647879976231365614463724856, 5.30568889524918928707048148403, 5.31422801123583529060166105933, 6.11259066250300813454781854460, 6.22044722763440522778596529484, 6.36106368773154639889763098411, 6.36133432961955851676320359289, 7.19303835531287636026552760439

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