Properties

Label 2-56e2-1.1-c1-0-30
Degree $2$
Conductor $3136$
Sign $1$
Analytic cond. $25.0410$
Root an. cond. $5.00410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 3·9-s + 4·11-s + 2·13-s + 6·17-s + 8·19-s − 25-s − 6·29-s − 8·31-s + 2·37-s − 2·41-s + 4·43-s − 6·45-s + 8·47-s − 6·53-s + 8·55-s − 6·61-s + 4·65-s + 4·67-s − 8·71-s − 10·73-s + 16·79-s + 9·81-s + 8·83-s + 12·85-s + 6·89-s + 16·95-s + ⋯
L(s)  = 1  + 0.894·5-s − 9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 0.894·45-s + 1.16·47-s − 0.824·53-s + 1.07·55-s − 0.768·61-s + 0.496·65-s + 0.488·67-s − 0.949·71-s − 1.17·73-s + 1.80·79-s + 81-s + 0.878·83-s + 1.30·85-s + 0.635·89-s + 1.64·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.0410\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3136} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.454753303\)
\(L(\frac12)\) \(\approx\) \(2.454753303\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.065821423802092395362181435296, −7.81893808816424475924755874673, −7.31356679414241138388846467495, −6.08808818165492598018676693090, −5.81038841174965326632002876390, −5.10093822894527537496894350174, −3.70152283742019411667382089257, −3.22234166202046004358507312699, −1.93975903006317803281774950018, −1.01536704926852438245703233735, 1.01536704926852438245703233735, 1.93975903006317803281774950018, 3.22234166202046004358507312699, 3.70152283742019411667382089257, 5.10093822894527537496894350174, 5.81038841174965326632002876390, 6.08808818165492598018676693090, 7.31356679414241138388846467495, 7.81893808816424475924755874673, 9.065821423802092395362181435296

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