Properties

Label 2-5054-1.1-c1-0-10
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 − 3.16·3-s + 4-s − 2.51·5-s + 3.16·6-s − 7-s − 8-s + 7.01·9-s + 2.51·10-s − 2.61·11-s − 3.16·12-s + 4.82·13-s + 14-s + 7.94·15-s + 16-s − 4.26·17-s − 7.01·18-s − 2.51·20-s + 3.16·21-s + 2.61·22-s − 7.62·23-s + 3.16·24-s + 1.30·25-s − 4.82·26-s − 12.6·27-s − 28-s + 3.60·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.82·3-s + 0.5·4-s − 1.12·5-s + 1.29·6-s − 0.377·7-s − 0.353·8-s + 2.33·9-s + 0.793·10-s − 0.787·11-s − 0.913·12-s + 1.33·13-s + 0.267·14-s + 2.05·15-s + 0.250·16-s − 1.03·17-s − 1.65·18-s − 0.561·20-s + 0.690·21-s + 0.557·22-s − 1.59·23-s + 0.645·24-s + 0.260·25-s − 0.946·26-s − 2.44·27-s − 0.188·28-s + 0.670·29-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5054,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2168455602\)
\(L(\frac12)\) \(\approx\) \(0.2168455602\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.16T + 3T^{2} \)
5 \( 1 + 2.51T + 5T^{2} \)
11 \( 1 + 2.61T + 11T^{2} \)
13 \( 1 - 4.82T + 13T^{2} \)
17 \( 1 + 4.26T + 17T^{2} \)
23 \( 1 + 7.62T + 23T^{2} \)
29 \( 1 - 3.60T + 29T^{2} \)
31 \( 1 - 7.24T + 31T^{2} \)
37 \( 1 - 3.23T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 4.22T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 7.34T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 2.33T + 61T^{2} \)
67 \( 1 + 8.50T + 67T^{2} \)
71 \( 1 + 4.76T + 71T^{2} \)
73 \( 1 + 1.27T + 73T^{2} \)
79 \( 1 + 4.48T + 79T^{2} \)
83 \( 1 + 5.10T + 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−8.037248741654419401848613962262, −7.57825880248832520532566603763, −6.64574585380130263009075034308, −6.17644457319470256111612228093, −5.61499068754995397309463645076, −4.36684496034791477280631742349, −4.14344575024315214612921195926, −2.77251448771560041000138754116, −1.35828877105443320653201601957, −0.33651256004019389585828036098, 0.33651256004019389585828036098, 1.35828877105443320653201601957, 2.77251448771560041000138754116, 4.14344575024315214612921195926, 4.36684496034791477280631742349, 5.61499068754995397309463645076, 6.17644457319470256111612228093, 6.64574585380130263009075034308, 7.57825880248832520532566603763, 8.037248741654419401848613962262

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