Properties

Label 2-5148-1.1-c1-0-40
Degree $2$
Conductor $5148$
Sign $-1$
Analytic cond. $41.1069$
Root an. cond. $6.41147$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.51·5-s − 1.32·7-s − 11-s − 13-s + 4.83·17-s + 5.32·19-s − 8.73·23-s − 2.70·25-s − 10.4·29-s + 0.485·31-s − 2·35-s + 2.64·37-s − 10.7·41-s − 0.193·43-s − 5.02·47-s − 5.25·49-s + 7.67·53-s − 1.51·55-s + 7.02·59-s + 6.25·61-s − 1.51·65-s + 9.18·67-s − 14.6·71-s + 10.9·73-s + 1.32·77-s − 16.5·79-s − 0.386·83-s + ⋯
L(s)  = 1  + 0.677·5-s − 0.499·7-s − 0.301·11-s − 0.277·13-s + 1.17·17-s + 1.22·19-s − 1.82·23-s − 0.541·25-s − 1.94·29-s + 0.0872·31-s − 0.338·35-s + 0.434·37-s − 1.67·41-s − 0.0294·43-s − 0.733·47-s − 0.750·49-s + 1.05·53-s − 0.204·55-s + 0.915·59-s + 0.800·61-s − 0.187·65-s + 1.12·67-s − 1.74·71-s + 1.28·73-s + 0.150·77-s − 1.85·79-s − 0.0424·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5148\)    =    \(2^{2} \cdot 3^{2} \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(41.1069\)
Root analytic conductor: \(6.41147\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5148,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + T \)
13 \( 1 + T \)
good5 \( 1 - 1.51T + 5T^{2} \)
7 \( 1 + 1.32T + 7T^{2} \)
17 \( 1 - 4.83T + 17T^{2} \)
19 \( 1 - 5.32T + 19T^{2} \)
23 \( 1 + 8.73T + 23T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 - 0.485T + 31T^{2} \)
37 \( 1 - 2.64T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 0.193T + 43T^{2} \)
47 \( 1 + 5.02T + 47T^{2} \)
53 \( 1 - 7.67T + 53T^{2} \)
59 \( 1 - 7.02T + 59T^{2} \)
61 \( 1 - 6.25T + 61T^{2} \)
67 \( 1 - 9.18T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 16.5T + 79T^{2} \)
83 \( 1 + 0.386T + 83T^{2} \)
89 \( 1 - 2.92T + 89T^{2} \)
97 \( 1 + 6.64T + 97T^{2} \)
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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

−7.79319189812867040256354231971, −7.23168192452761721750415034360, −6.30810852018867048429768809932, −5.58443159441178744156072904901, −5.25572378036782463297987118048, −3.94386404394763439714371274936, −3.35249954059876053758631797398, −2.32410615610219134338659763916, −1.47341358812872833137824423954, 0, 1.47341358812872833137824423954, 2.32410615610219134338659763916, 3.35249954059876053758631797398, 3.94386404394763439714371274936, 5.25572378036782463297987118048, 5.58443159441178744156072904901, 6.30810852018867048429768809932, 7.23168192452761721750415034360, 7.79319189812867040256354231971

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