Properties

Label 4-112e2-1.1-c13e2-0-1
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $14423.6$
Root an. cond. $10.9589$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.10e3·3-s + 7.55e4·5-s + 2.35e5·7-s − 2.96e5·9-s − 3.33e6·11-s + 7.99e6·13-s − 8.35e7·15-s + 3.90e7·17-s − 1.24e8·19-s − 2.60e8·21-s − 1.90e9·23-s + 1.95e9·25-s + 2.45e8·27-s − 2.45e8·29-s + 1.10e10·31-s + 3.68e9·33-s + 1.77e10·35-s − 3.96e10·37-s − 8.84e9·39-s − 2.43e9·41-s − 1.78e10·43-s − 2.24e10·45-s − 6.44e10·47-s + 4.15e10·49-s − 4.31e10·51-s − 1.26e11·53-s − 2.51e11·55-s + ⋯
L(s)  = 1  − 0.875·3-s + 2.16·5-s + 0.755·7-s − 0.186·9-s − 0.567·11-s + 0.459·13-s − 1.89·15-s + 0.392·17-s − 0.605·19-s − 0.662·21-s − 2.67·23-s + 1.60·25-s + 0.122·27-s − 0.0765·29-s + 2.22·31-s + 0.497·33-s + 1.63·35-s − 2.53·37-s − 0.402·39-s − 0.0799·41-s − 0.429·43-s − 0.402·45-s − 0.872·47-s + 3/7·49-s − 0.343·51-s − 0.783·53-s − 1.22·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(14423.6\)
Root analytic conductor: \(10.9589\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{112} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 12544,\ (\ :13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - p^{6} T )^{2} \)
good3$D_{4}$ \( 1 + 1106 T + 56290 p^{3} T^{2} + 1106 p^{13} T^{3} + p^{26} T^{4} \)
5$D_{4}$ \( 1 - 15106 p T + 749493058 p T^{2} - 15106 p^{14} T^{3} + p^{26} T^{4} \)
11$D_{4}$ \( 1 + 303260 p T - 17283060398 p^{3} T^{2} + 303260 p^{14} T^{3} + p^{26} T^{4} \)
13$D_{4}$ \( 1 - 7998102 T + 313454244594482 T^{2} - 7998102 p^{13} T^{3} + p^{26} T^{4} \)
17$D_{4}$ \( 1 - 39024832 T + 10050449815930430 T^{2} - 39024832 p^{13} T^{3} + p^{26} T^{4} \)
19$D_{4}$ \( 1 + 124092934 T + 64593685044636582 T^{2} + 124092934 p^{13} T^{3} + p^{26} T^{4} \)
23$D_{4}$ \( 1 + 1900816000 T + 1895455828613638766 T^{2} + 1900816000 p^{13} T^{3} + p^{26} T^{4} \)
29$D_{4}$ \( 1 + 245135152 T + 11525694953905661654 T^{2} + 245135152 p^{13} T^{3} + p^{26} T^{4} \)
31$D_{4}$ \( 1 - 11010560148 T + 72353550492368544158 T^{2} - 11010560148 p^{13} T^{3} + p^{26} T^{4} \)
37$D_{4}$ \( 1 + 39630491216 T + \)\(77\!\cdots\!58\)\( T^{2} + 39630491216 p^{13} T^{3} + p^{26} T^{4} \)
41$D_{4}$ \( 1 + 2431027368 T + \)\(18\!\cdots\!98\)\( T^{2} + 2431027368 p^{13} T^{3} + p^{26} T^{4} \)
43$D_{4}$ \( 1 + 17823138316 T + \)\(10\!\cdots\!50\)\( T^{2} + 17823138316 p^{13} T^{3} + p^{26} T^{4} \)
47$D_{4}$ \( 1 + 64488311076 T + \)\(22\!\cdots\!98\)\( T^{2} + 64488311076 p^{13} T^{3} + p^{26} T^{4} \)
53$D_{4}$ \( 1 + 126504176628 T + \)\(48\!\cdots\!42\)\( T^{2} + 126504176628 p^{13} T^{3} + p^{26} T^{4} \)
59$D_{4}$ \( 1 + 341259961238 T + \)\(17\!\cdots\!94\)\( T^{2} + 341259961238 p^{13} T^{3} + p^{26} T^{4} \)
61$D_{4}$ \( 1 - 447240700746 T + \)\(34\!\cdots\!66\)\( T^{2} - 447240700746 p^{13} T^{3} + p^{26} T^{4} \)
67$D_{4}$ \( 1 - 2071322290888 T + \)\(21\!\cdots\!10\)\( T^{2} - 2071322290888 p^{13} T^{3} + p^{26} T^{4} \)
71$D_{4}$ \( 1 + 650434465720 T + \)\(22\!\cdots\!22\)\( T^{2} + 650434465720 p^{13} T^{3} + p^{26} T^{4} \)
73$D_{4}$ \( 1 + 1449809330116 T + \)\(32\!\cdots\!30\)\( T^{2} + 1449809330116 p^{13} T^{3} + p^{26} T^{4} \)
79$D_{4}$ \( 1 + 1525152397656 T + \)\(78\!\cdots\!62\)\( T^{2} + 1525152397656 p^{13} T^{3} + p^{26} T^{4} \)
83$D_{4}$ \( 1 + 4257517639438 T + \)\(13\!\cdots\!62\)\( T^{2} + 4257517639438 p^{13} T^{3} + p^{26} T^{4} \)
89$D_{4}$ \( 1 - 12593651222628 T + \)\(83\!\cdots\!34\)\( T^{2} - 12593651222628 p^{13} T^{3} + p^{26} T^{4} \)
97$D_{4}$ \( 1 + 16541570007760 T + \)\(16\!\cdots\!54\)\( T^{2} + 16541570007760 p^{13} T^{3} + p^{26} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.52735526343663658879478298237, −10.43636308371188668276749683364, −9.895986773143728070197857450411, −9.590511930215736587637983210787, −8.651230082603260870496737703494, −8.248158164146341286592110337254, −7.81045948803328565627167128847, −6.68925067399321625785393528110, −6.32218630871897498402665946143, −5.94636534997096971076325470581, −5.32176815379104257494490086985, −5.21858288214917127980143849416, −4.31382208032391867289151624906, −3.59497718335619403847167543334, −2.57040790386398871740614616408, −2.18310932298262035891072881319, −1.57919590120533796283628454942, −1.27300178612086040082084020465, 0, 0, 1.27300178612086040082084020465, 1.57919590120533796283628454942, 2.18310932298262035891072881319, 2.57040790386398871740614616408, 3.59497718335619403847167543334, 4.31382208032391867289151624906, 5.21858288214917127980143849416, 5.32176815379104257494490086985, 5.94636534997096971076325470581, 6.32218630871897498402665946143, 6.68925067399321625785393528110, 7.81045948803328565627167128847, 8.248158164146341286592110337254, 8.651230082603260870496737703494, 9.590511930215736587637983210787, 9.895986773143728070197857450411, 10.43636308371188668276749683364, 10.52735526343663658879478298237

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