Properties

Label 2-7938-1.1-c1-0-55
Degree $2$
Conductor $7938$
Sign $1$
Analytic cond. $63.3852$
Root an. cond. $7.96148$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 0.267·5-s + 8-s + 0.267·10-s − 6.19·11-s + 6.46·13-s + 16-s + 7·17-s − 0.732·19-s + 0.267·20-s − 6.19·22-s + 4.19·23-s − 4.92·25-s + 6.46·26-s − 1.53·29-s − 8.19·31-s + 32-s + 7·34-s + 10.6·37-s − 0.732·38-s + 0.267·40-s + 2.53·41-s − 1.46·43-s − 6.19·44-s + 4.19·46-s + 4.73·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.119·5-s + 0.353·8-s + 0.0847·10-s − 1.86·11-s + 1.79·13-s + 0.250·16-s + 1.69·17-s − 0.167·19-s + 0.0599·20-s − 1.32·22-s + 0.874·23-s − 0.985·25-s + 1.26·26-s − 0.285·29-s − 1.47·31-s + 0.176·32-s + 1.20·34-s + 1.75·37-s − 0.118·38-s + 0.0423·40-s + 0.396·41-s − 0.223·43-s − 0.934·44-s + 0.618·46-s + 0.690·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7938\)    =    \(2 \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(63.3852\)
Root analytic conductor: \(7.96148\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7938,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.493765844\)
\(L(\frac12)\) \(\approx\) \(3.493765844\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 0.267T + 5T^{2} \)
11 \( 1 + 6.19T + 11T^{2} \)
13 \( 1 - 6.46T + 13T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 + 0.732T + 19T^{2} \)
23 \( 1 - 4.19T + 23T^{2} \)
29 \( 1 + 1.53T + 29T^{2} \)
31 \( 1 + 8.19T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 - 2.53T + 41T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 - 9.46T + 53T^{2} \)
59 \( 1 - 4.19T + 59T^{2} \)
61 \( 1 + 3.92T + 61T^{2} \)
67 \( 1 + 6.73T + 67T^{2} \)
71 \( 1 + 6.53T + 71T^{2} \)
73 \( 1 + 8.26T + 73T^{2} \)
79 \( 1 + 9.12T + 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 - 9.92T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−7.63154737740895890823531401902, −7.35062694648888552997509807419, −6.05193986313449556641568717912, −5.77623355945256934844047712964, −5.21927104765924035335365372707, −4.24528162004751063292812752860, −3.47672204224157539676236419226, −2.88682126596248651545546716194, −1.91446519504405065760906291454, −0.846087250399229237515631728529, 0.846087250399229237515631728529, 1.91446519504405065760906291454, 2.88682126596248651545546716194, 3.47672204224157539676236419226, 4.24528162004751063292812752860, 5.21927104765924035335365372707, 5.77623355945256934844047712964, 6.05193986313449556641568717912, 7.35062694648888552997509807419, 7.63154737740895890823531401902

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