Properties

Label 2-5054-1.1-c1-0-111
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 + 2.40·3-s + 4-s + 1.76·5-s + 2.40·6-s − 7-s + 8-s + 2.79·9-s + 1.76·10-s + 1.18·11-s + 2.40·12-s + 0.361·13-s − 14-s + 4.24·15-s + 16-s + 3.54·17-s + 2.79·18-s + 1.76·20-s − 2.40·21-s + 1.18·22-s − 2.87·23-s + 2.40·24-s − 1.89·25-s + 0.361·26-s − 0.489·27-s − 28-s + 5.10·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.39·3-s + 0.5·4-s + 0.787·5-s + 0.982·6-s − 0.377·7-s + 0.353·8-s + 0.932·9-s + 0.557·10-s + 0.357·11-s + 0.695·12-s + 0.100·13-s − 0.267·14-s + 1.09·15-s + 0.250·16-s + 0.859·17-s + 0.659·18-s + 0.393·20-s − 0.525·21-s + 0.252·22-s − 0.599·23-s + 0.491·24-s − 0.379·25-s + 0.0709·26-s − 0.0942·27-s − 0.188·28-s + 0.947·29-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

\(L(1)\) \(\approx\) \(6.180431306\)
\(L(\frac12)\) \(\approx\) \(6.180431306\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.40T + 3T^{2} \)
5 \( 1 - 1.76T + 5T^{2} \)
11 \( 1 - 1.18T + 11T^{2} \)
13 \( 1 - 0.361T + 13T^{2} \)
17 \( 1 - 3.54T + 17T^{2} \)
23 \( 1 + 2.87T + 23T^{2} \)
29 \( 1 - 5.10T + 29T^{2} \)
31 \( 1 + 1.38T + 31T^{2} \)
37 \( 1 + 3.35T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 3.47T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 + 7.72T + 59T^{2} \)
61 \( 1 + 1.99T + 61T^{2} \)
67 \( 1 - 8.30T + 67T^{2} \)
71 \( 1 + 0.226T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 - 5.28T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 1.87T + 89T^{2} \)
97 \( 1 + 3.99T + 97T^{2} \)
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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

−8.142850612621540918653527364474, −7.58327614551130216997300396148, −6.78964246321469075775159659258, −5.94087193581048102976044554088, −5.45759383983685373476744865716, −4.20612792526149849465646620532, −3.73269111248161254977692109573, −2.78484396637408625749808664576, −2.30013285996831195968224095864, −1.25753981066687820218593731211, 1.25753981066687820218593731211, 2.30013285996831195968224095864, 2.78484396637408625749808664576, 3.73269111248161254977692109573, 4.20612792526149849465646620532, 5.45759383983685373476744865716, 5.94087193581048102976044554088, 6.78964246321469075775159659258, 7.58327614551130216997300396148, 8.142850612621540918653527364474

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