Properties

Label 4-112e2-1.1-c3e2-0-2
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $43.6684$
Root an. cond. $2.57064$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·3-s − 7·5-s − 28·7-s + 27·9-s − 5·11-s − 28·13-s − 49·15-s + 21·17-s + 49·19-s − 196·21-s − 159·23-s + 125·25-s + 224·27-s + 116·29-s + 147·31-s − 35·33-s + 196·35-s − 219·37-s − 196·39-s + 700·41-s + 248·43-s − 189·45-s + 525·47-s + 441·49-s + 147·51-s − 303·53-s + 35·55-s + ⋯
L(s)  = 1  + 1.34·3-s − 0.626·5-s − 1.51·7-s + 9-s − 0.137·11-s − 0.597·13-s − 0.843·15-s + 0.299·17-s + 0.591·19-s − 2.03·21-s − 1.44·23-s + 25-s + 1.59·27-s + 0.742·29-s + 0.851·31-s − 0.184·33-s + 0.946·35-s − 0.973·37-s − 0.804·39-s + 2.66·41-s + 0.879·43-s − 0.626·45-s + 1.62·47-s + 9/7·49-s + 0.403·51-s − 0.785·53-s + 0.0858·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(43.6684\)
Root analytic conductor: \(2.57064\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{112} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.250252614\)
\(L(\frac12)\) \(\approx\) \(2.250252614\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + 4 p T + p^{3} T^{2} \)
good3$C_2^2$ \( 1 - 7 T + 22 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 7 T - 76 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 5 T - 1306 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 14 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 21 T - 4472 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 49 T - 4458 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 159 T + 13114 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 147 T - 8182 T^{2} - 147 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 219 T - 2692 T^{2} + 219 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 350 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 124 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 525 T + 171802 T^{2} - 525 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 303 T - 57068 T^{2} + 303 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 105 T - 194354 T^{2} + 105 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 413 T - 56412 T^{2} - 413 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 415 T - 128538 T^{2} - 415 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 432 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 1113 T + 849752 T^{2} - 1113 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 103 T - 482430 T^{2} + 103 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 1092 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 329 T - 596728 T^{2} - 329 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 882 T + p^{3} T^{2} )^{2} \)
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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

−13.77522464107123319740429387468, −12.60229221999992733465430575110, −12.46178875769074656305838918373, −12.32268613845186229640532117117, −11.29339011735704643568182601843, −10.59689678225756320791338750001, −10.11391122182318116727276009076, −9.424324307334658082833237055529, −9.363292393454464971890291657794, −8.371762140282821416618044941954, −8.144992341126555120023899730488, −7.37199801787487048333450219664, −6.90142190842878273403837651704, −6.22875899810267257356607438793, −5.35880471342692311613737537011, −4.28069863714865627094678292610, −3.79360406178870572273987475420, −2.83517984122365318093184997326, −2.59600639105346831325938855419, −0.76674519542290344649263439168, 0.76674519542290344649263439168, 2.59600639105346831325938855419, 2.83517984122365318093184997326, 3.79360406178870572273987475420, 4.28069863714865627094678292610, 5.35880471342692311613737537011, 6.22875899810267257356607438793, 6.90142190842878273403837651704, 7.37199801787487048333450219664, 8.144992341126555120023899730488, 8.371762140282821416618044941954, 9.363292393454464971890291657794, 9.424324307334658082833237055529, 10.11391122182318116727276009076, 10.59689678225756320791338750001, 11.29339011735704643568182601843, 12.32268613845186229640532117117, 12.46178875769074656305838918373, 12.60229221999992733465430575110, 13.77522464107123319740429387468

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