Properties

Label 2-5054-1.1-c1-0-39
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 − 0.618·3-s + 4-s − 0.381·5-s + 0.618·6-s + 7-s − 8-s − 2.61·9-s + 0.381·10-s + 1.85·11-s − 0.618·12-s + 4.47·13-s − 14-s + 0.236·15-s + 16-s + 5.23·17-s + 2.61·18-s − 0.381·20-s − 0.618·21-s − 1.85·22-s − 8.94·23-s + 0.618·24-s − 4.85·25-s − 4.47·26-s + 3.47·27-s + 28-s + 6.85·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.356·3-s + 0.5·4-s − 0.170·5-s + 0.252·6-s + 0.377·7-s − 0.353·8-s − 0.872·9-s + 0.120·10-s + 0.559·11-s − 0.178·12-s + 1.24·13-s − 0.267·14-s + 0.0609·15-s + 0.250·16-s + 1.26·17-s + 0.617·18-s − 0.0854·20-s − 0.134·21-s − 0.395·22-s − 1.86·23-s + 0.126·24-s − 0.970·25-s − 0.877·26-s + 0.668·27-s + 0.188·28-s + 1.27·29-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

\(L(1)\) \(\approx\) \(1.216021536\)
\(L(\frac12)\) \(\approx\) \(1.216021536\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 3T^{2} \)
5 \( 1 + 0.381T + 5T^{2} \)
11 \( 1 - 1.85T + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
23 \( 1 + 8.94T + 23T^{2} \)
29 \( 1 - 6.85T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 1.14T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 2.85T + 43T^{2} \)
47 \( 1 + 4.14T + 47T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 - 8.56T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 0.763T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 - 6.76T + 83T^{2} \)
89 \( 1 + 9.85T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−8.162646111900772498179998260656, −7.88725589967397767720685208378, −6.73566626773931879809450467347, −5.99273230443321238428187674388, −5.72674116193616171270915631150, −4.47951579912647011050824795580, −3.67164167160980973347791700435, −2.78165750864897515899034714034, −1.62953851181612196384905410238, −0.71292337534387395828772496703, 0.71292337534387395828772496703, 1.62953851181612196384905410238, 2.78165750864897515899034714034, 3.67164167160980973347791700435, 4.47951579912647011050824795580, 5.72674116193616171270915631150, 5.99273230443321238428187674388, 6.73566626773931879809450467347, 7.88725589967397767720685208378, 8.162646111900772498179998260656

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