Properties

Label 2-3744-1.1-c1-0-26
Degree $2$
Conductor $3744$
Sign $1$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
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  + 3.56·5-s + 3.56·7-s − 2·11-s − 13-s − 3.56·17-s + 6·19-s + 7.68·25-s − 8.24·29-s − 1.12·31-s + 12.6·35-s + 2.68·37-s + 1.12·41-s + 11.8·43-s + 10.6·47-s + 5.68·49-s + 13.1·53-s − 7.12·55-s + 6·59-s − 11.3·61-s − 3.56·65-s + 6·67-s − 10.6·71-s + 10·73-s − 7.12·77-s + 12·79-s − 7.36·83-s − 12.6·85-s + ⋯
L(s)  = 1  + 1.59·5-s + 1.34·7-s − 0.603·11-s − 0.277·13-s − 0.863·17-s + 1.37·19-s + 1.53·25-s − 1.53·29-s − 0.201·31-s + 2.14·35-s + 0.441·37-s + 0.175·41-s + 1.80·43-s + 1.55·47-s + 0.812·49-s + 1.80·53-s − 0.960·55-s + 0.781·59-s − 1.45·61-s − 0.441·65-s + 0.733·67-s − 1.26·71-s + 1.17·73-s − 0.811·77-s + 1.35·79-s − 0.808·83-s − 1.37·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.063848906\)
\(L(\frac12)\) \(\approx\) \(3.063848906\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
good5 \( 1 - 3.56T + 5T^{2} \)
7 \( 1 - 3.56T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 + 3.56T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 + 1.12T + 31T^{2} \)
37 \( 1 - 2.68T + 37T^{2} \)
41 \( 1 - 1.12T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 6T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 7.36T + 83T^{2} \)
89 \( 1 + 8.24T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−8.672379834228386912156012530942, −7.60657405290198943783752617810, −7.24589121841923659251995738959, −6.02461502923621686825853382799, −5.49849085553364673524752475206, −4.98313064299661300307884445325, −4.01335898735758249287994987731, −2.58315846129967278843404192833, −2.08638876056525702089597292242, −1.09232464334491493511682723071, 1.09232464334491493511682723071, 2.08638876056525702089597292242, 2.58315846129967278843404192833, 4.01335898735758249287994987731, 4.98313064299661300307884445325, 5.49849085553364673524752475206, 6.02461502923621686825853382799, 7.24589121841923659251995738959, 7.60657405290198943783752617810, 8.672379834228386912156012530942

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