Properties

Label 2-5239-1.1-c1-0-87
Degree $2$
Conductor $5239$
Sign $1$
Analytic cond. $41.8336$
Root an. cond. $6.46789$
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  + 1.88·2-s − 0.327·3-s + 1.57·4-s − 1.75·5-s − 0.619·6-s − 0.652·7-s − 0.809·8-s − 2.89·9-s − 3.32·10-s − 4.04·11-s − 0.515·12-s − 1.23·14-s + 0.576·15-s − 4.67·16-s + 2.05·17-s − 5.46·18-s + 7.52·19-s − 2.76·20-s + 0.213·21-s − 7.63·22-s + 1.17·23-s + 0.265·24-s − 1.90·25-s + 1.93·27-s − 1.02·28-s + 6.42·29-s + 1.08·30-s + ⋯
L(s)  = 1  + 1.33·2-s − 0.189·3-s + 0.785·4-s − 0.786·5-s − 0.253·6-s − 0.246·7-s − 0.286·8-s − 0.964·9-s − 1.05·10-s − 1.21·11-s − 0.148·12-s − 0.329·14-s + 0.148·15-s − 1.16·16-s + 0.497·17-s − 1.28·18-s + 1.72·19-s − 0.617·20-s + 0.0466·21-s − 1.62·22-s + 0.245·23-s + 0.0542·24-s − 0.381·25-s + 0.371·27-s − 0.193·28-s + 1.19·29-s + 0.198·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5239\)    =    \(13^{2} \cdot 31\)
Sign: $1$
Analytic conductor: \(41.8336\)
Root analytic conductor: \(6.46789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5239,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.955540587\)
\(L(\frac12)\) \(\approx\) \(1.955540587\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 + T \)
good2 \( 1 - 1.88T + 2T^{2} \)
3 \( 1 + 0.327T + 3T^{2} \)
5 \( 1 + 1.75T + 5T^{2} \)
7 \( 1 + 0.652T + 7T^{2} \)
11 \( 1 + 4.04T + 11T^{2} \)
17 \( 1 - 2.05T + 17T^{2} \)
19 \( 1 - 7.52T + 19T^{2} \)
23 \( 1 - 1.17T + 23T^{2} \)
29 \( 1 - 6.42T + 29T^{2} \)
37 \( 1 - 4.69T + 37T^{2} \)
41 \( 1 + 5.09T + 41T^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 - 1.20T + 47T^{2} \)
53 \( 1 - 0.419T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 6.53T + 71T^{2} \)
73 \( 1 + 0.170T + 73T^{2} \)
79 \( 1 - 8.39T + 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 - 7.61T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−7.969895727154218973374516401128, −7.49245734056295059042697582395, −6.49232373552786930456378774246, −5.83037397829064213740703746472, −5.10739642563363994713285053063, −4.77899108418129970396276069633, −3.51276536482085612720609854932, −3.23664443530676374388269830860, −2.40634543284122731315505440722, −0.59234930320095958156472998342, 0.59234930320095958156472998342, 2.40634543284122731315505440722, 3.23664443530676374388269830860, 3.51276536482085612720609854932, 4.77899108418129970396276069633, 5.10739642563363994713285053063, 5.83037397829064213740703746472, 6.49232373552786930456378774246, 7.49245734056295059042697582395, 7.969895727154218973374516401128

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