Properties

Label 2-5054-1.1-c1-0-139
Degree $2$
Conductor $5054$
Sign $1$
Analytic cond. $40.3563$
Root an. cond. $6.35266$
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 + 3.21·3-s + 4-s + 1.59·5-s + 3.21·6-s + 7-s + 8-s + 7.34·9-s + 1.59·10-s − 1.21·11-s + 3.21·12-s + 4.71·13-s + 14-s + 5.12·15-s + 16-s − 7.55·17-s + 7.34·18-s + 1.59·20-s + 3.21·21-s − 1.21·22-s − 6.52·23-s + 3.21·24-s − 2.45·25-s + 4.71·26-s + 13.9·27-s + 28-s + 0.118·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.85·3-s + 0.5·4-s + 0.712·5-s + 1.31·6-s + 0.377·7-s + 0.353·8-s + 2.44·9-s + 0.504·10-s − 0.366·11-s + 0.928·12-s + 1.30·13-s + 0.267·14-s + 1.32·15-s + 0.250·16-s − 1.83·17-s + 1.73·18-s + 0.356·20-s + 0.701·21-s − 0.259·22-s − 1.36·23-s + 0.656·24-s − 0.491·25-s + 0.925·26-s + 2.68·27-s + 0.188·28-s + 0.0219·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5054\)    =    \(2 \cdot 7 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(40.3563\)
Root analytic conductor: \(6.35266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5054} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.611499094\)
\(L(\frac12)\) \(\approx\) \(7.611499094\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
19 \( 1 \)
good3 \( 1 - 3.21T + 3T^{2} \)
5 \( 1 - 1.59T + 5T^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 - 4.71T + 13T^{2} \)
17 \( 1 + 7.55T + 17T^{2} \)
23 \( 1 + 6.52T + 23T^{2} \)
29 \( 1 - 0.118T + 29T^{2} \)
31 \( 1 - 6.31T + 31T^{2} \)
37 \( 1 + 5.18T + 37T^{2} \)
41 \( 1 - 7.21T + 41T^{2} \)
43 \( 1 + 0.313T + 43T^{2} \)
47 \( 1 - 0.118T + 47T^{2} \)
53 \( 1 + 2.31T + 53T^{2} \)
59 \( 1 - 1.25T + 59T^{2} \)
61 \( 1 - 7.84T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 4.97T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + 6.25T + 79T^{2} \)
83 \( 1 + 9.99T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 - 7.64T + 97T^{2} \)
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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

−8.336209882608231046459090813148, −7.67770686690285324612760720698, −6.72597933436836848514190464270, −6.20667008815115076374071074350, −5.17231846792791669759757806411, −4.11124825826309638269111578565, −3.90252844578668575321341139225, −2.70575034786379614755575302960, −2.21762957557922316401703156732, −1.47712610330463286779853308799, 1.47712610330463286779853308799, 2.21762957557922316401703156732, 2.70575034786379614755575302960, 3.90252844578668575321341139225, 4.11124825826309638269111578565, 5.17231846792791669759757806411, 6.20667008815115076374071074350, 6.72597933436836848514190464270, 7.67770686690285324612760720698, 8.336209882608231046459090813148

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