Properties

Label 2-56e2-1.1-c1-0-42
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  + 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: \(0\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.039410017\)
\(L(\frac12)\) \(\approx\) \(4.039410017\)
\(L(\frac{3}{2})\) 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

−8.613387530453077204772370730115, −8.031996104849329823930918913127, −7.41222969775341712005479565068, −6.69932322919898823116211066736, −5.52190697314800540673323359415, −4.74391206643302649394539653497, −3.67640048216274307516056199890, −2.98885024194690194695850849492, −2.28746912531373009454400424430, −1.23644851964910009262807684617, 1.23644851964910009262807684617, 2.28746912531373009454400424430, 2.98885024194690194695850849492, 3.67640048216274307516056199890, 4.74391206643302649394539653497, 5.52190697314800540673323359415, 6.69932322919898823116211066736, 7.41222969775341712005479565068, 8.031996104849329823930918913127, 8.613387530453077204772370730115

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