Properties

Label 2-56e2-1.1-c1-0-31
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5-s + 6·9-s − 11-s + 2·13-s + 3·15-s − 3·17-s − 5·19-s + 3·23-s − 4·25-s − 9·27-s + 6·29-s − 31-s + 3·33-s + 5·37-s − 6·39-s + 10·41-s − 4·43-s − 6·45-s + 47-s + 9·51-s + 9·53-s + 55-s + 15·57-s − 3·59-s + 3·61-s − 2·65-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 2·9-s − 0.301·11-s + 0.554·13-s + 0.774·15-s − 0.727·17-s − 1.14·19-s + 0.625·23-s − 4/5·25-s − 1.73·27-s + 1.11·29-s − 0.179·31-s + 0.522·33-s + 0.821·37-s − 0.960·39-s + 1.56·41-s − 0.609·43-s − 0.894·45-s + 0.145·47-s + 1.26·51-s + 1.23·53-s + 0.134·55-s + 1.98·57-s − 0.390·59-s + 0.384·61-s − 0.248·65-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: \(1\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−8.230315165473213095839519633705, −7.38315583202405666291782015581, −6.56899809479189996756800142405, −6.11960412337872958718479712327, −5.29913325153166278991319869627, −4.49040947012230363312554161259, −3.93566966697450588624507732655, −2.44876208464128486413690607456, −1.08713788900686923431368242918, 0, 1.08713788900686923431368242918, 2.44876208464128486413690607456, 3.93566966697450588624507732655, 4.49040947012230363312554161259, 5.29913325153166278991319869627, 6.11960412337872958718479712327, 6.56899809479189996756800142405, 7.38315583202405666291782015581, 8.230315165473213095839519633705

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