Properties

Label 2-5054-1.1-c1-0-58
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 + 0.806·3-s + 4-s − 1.86·5-s + 0.806·6-s − 7-s + 8-s − 2.35·9-s − 1.86·10-s + 5.76·11-s + 0.806·12-s − 2.28·13-s − 14-s − 1.50·15-s + 16-s + 5.11·17-s − 2.35·18-s − 1.86·20-s − 0.806·21-s + 5.76·22-s + 5.31·23-s + 0.806·24-s − 1.50·25-s − 2.28·26-s − 4.31·27-s − 28-s − 6.54·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.465·3-s + 0.5·4-s − 0.835·5-s + 0.329·6-s − 0.377·7-s + 0.353·8-s − 0.783·9-s − 0.591·10-s + 1.73·11-s + 0.232·12-s − 0.634·13-s − 0.267·14-s − 0.388·15-s + 0.250·16-s + 1.24·17-s − 0.553·18-s − 0.417·20-s − 0.175·21-s + 1.22·22-s + 1.10·23-s + 0.164·24-s − 0.301·25-s − 0.448·26-s − 0.829·27-s − 0.188·28-s − 1.21·29-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

\(L(1)\) \(\approx\) \(3.084172842\)
\(L(\frac12)\) \(\approx\) \(3.084172842\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.806T + 3T^{2} \)
5 \( 1 + 1.86T + 5T^{2} \)
11 \( 1 - 5.76T + 11T^{2} \)
13 \( 1 + 2.28T + 13T^{2} \)
17 \( 1 - 5.11T + 17T^{2} \)
23 \( 1 - 5.31T + 23T^{2} \)
29 \( 1 + 6.54T + 29T^{2} \)
31 \( 1 - 2.54T + 31T^{2} \)
37 \( 1 - 0.962T + 37T^{2} \)
41 \( 1 - 2.15T + 41T^{2} \)
43 \( 1 + 8.54T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 8.46T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 8.51T + 61T^{2} \)
67 \( 1 + 4.64T + 67T^{2} \)
71 \( 1 - 16.1T + 71T^{2} \)
73 \( 1 + 9.53T + 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + 1.86T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 6.54T + 97T^{2} \)
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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

−8.188712751720410036684350613074, −7.33821693483170485009727404504, −6.93613941078685031734641800289, −5.94737151681046574973227518749, −5.36816055063937679860177668592, −4.29647589315661998331982190354, −3.65007969008478237679611075959, −3.19030594728731679607591535558, −2.14624289452501128361044655970, −0.847092979050786792235401678512, 0.847092979050786792235401678512, 2.14624289452501128361044655970, 3.19030594728731679607591535558, 3.65007969008478237679611075959, 4.29647589315661998331982190354, 5.36816055063937679860177668592, 5.94737151681046574973227518749, 6.93613941078685031734641800289, 7.33821693483170485009727404504, 8.188712751720410036684350613074

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