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Properties

Label 2-56e2-1.1-c1-0-40
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  − 4·5-s − 3·9-s + 4·13-s + 8·17-s + 11·25-s − 10·29-s − 2·37-s + 8·41-s + 12·45-s − 14·53-s + 12·61-s − 16·65-s − 16·73-s + 9·81-s − 32·85-s − 16·89-s − 8·97-s − 20·101-s + 6·109-s − 14·113-s − 12·117-s + ⋯
L(s)  = 1  − 1.78·5-s − 9-s + 1.10·13-s + 1.94·17-s + 11/5·25-s − 1.85·29-s − 0.328·37-s + 1.24·41-s + 1.78·45-s − 1.92·53-s + 1.53·61-s − 1.98·65-s − 1.87·73-s + 81-s − 3.47·85-s − 1.69·89-s − 0.812·97-s − 1.99·101-s + 0.574·109-s − 1.31·113-s − 1.10·117-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: Trivial
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 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.123232783538407768276819081286, −7.79330766917692083191216207854, −7.01225284817552731900486031621, −5.90888986258636045999353273184, −5.33692971582806447373190589377, −4.14822368733080278963502028328, −3.55862068772781357146691102524, −2.94631965268932606563564149438, −1.23192402066262049129970636313, 0, 1.23192402066262049129970636313, 2.94631965268932606563564149438, 3.55862068772781357146691102524, 4.14822368733080278963502028328, 5.33692971582806447373190589377, 5.90888986258636045999353273184, 7.01225284817552731900486031621, 7.79330766917692083191216207854, 8.123232783538407768276819081286

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