Properties

Label 2-7488-1.1-c1-0-79
Degree $2$
Conductor $7488$
Sign $-1$
Analytic cond. $59.7919$
Root an. cond. $7.73252$
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  − 3.70·5-s + 4.20·7-s − 1.09·11-s − 13-s − 0.298·17-s − 1.09·19-s + 8.70·25-s − 2·29-s − 5.13·31-s − 15.5·35-s + 3.70·37-s + 9.40·41-s + 5.29·43-s − 4.20·47-s + 10.7·49-s + 1.40·53-s + 4.04·55-s − 13.5·59-s − 9.40·61-s + 3.70·65-s + 11.3·67-s − 8.25·71-s − 6·73-s − 4.59·77-s − 14.6·79-s + 7.32·83-s + 1.10·85-s + ⋯
L(s)  = 1  − 1.65·5-s + 1.59·7-s − 0.329·11-s − 0.277·13-s − 0.0723·17-s − 0.250·19-s + 1.74·25-s − 0.371·29-s − 0.922·31-s − 2.63·35-s + 0.608·37-s + 1.46·41-s + 0.808·43-s − 0.613·47-s + 1.52·49-s + 0.192·53-s + 0.545·55-s − 1.76·59-s − 1.20·61-s + 0.459·65-s + 1.38·67-s − 0.979·71-s − 0.702·73-s − 0.523·77-s − 1.64·79-s + 0.803·83-s + 0.119·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7488\)    =    \(2^{6} \cdot 3^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(59.7919\)
Root analytic conductor: \(7.73252\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7488,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good5 \( 1 + 3.70T + 5T^{2} \)
7 \( 1 - 4.20T + 7T^{2} \)
11 \( 1 + 1.09T + 11T^{2} \)
17 \( 1 + 0.298T + 17T^{2} \)
19 \( 1 + 1.09T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 5.13T + 31T^{2} \)
37 \( 1 - 3.70T + 37T^{2} \)
41 \( 1 - 9.40T + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 + 4.20T + 47T^{2} \)
53 \( 1 - 1.40T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 + 9.40T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 8.25T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 - 7.32T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 8.80T + 97T^{2} \)
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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

−7.68330143453315020917853757782, −7.24653683002995528832620308306, −6.12029391731427315789513145521, −5.24047437668904746894271690896, −4.52179966955378123593433680420, −4.17230983459498159093735245994, −3.22828908652212673182644652733, −2.24788638689645544235796437667, −1.18695215174687380244086291652, 0, 1.18695215174687380244086291652, 2.24788638689645544235796437667, 3.22828908652212673182644652733, 4.17230983459498159093735245994, 4.52179966955378123593433680420, 5.24047437668904746894271690896, 6.12029391731427315789513145521, 7.24653683002995528832620308306, 7.68330143453315020917853757782

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