Properties

Label 2-5239-1.1-c1-0-192
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  + 2.67·2-s − 1.67·3-s + 5.15·4-s + 2.86·5-s − 4.46·6-s − 5.21·7-s + 8.44·8-s − 0.209·9-s + 7.66·10-s + 0.107·11-s − 8.61·12-s − 13.9·14-s − 4.78·15-s + 12.2·16-s + 1.65·17-s − 0.560·18-s + 2.74·19-s + 14.7·20-s + 8.70·21-s + 0.287·22-s − 2.05·23-s − 14.1·24-s + 3.21·25-s + 5.36·27-s − 26.8·28-s + 8.54·29-s − 12.8·30-s + ⋯
L(s)  = 1  + 1.89·2-s − 0.964·3-s + 2.57·4-s + 1.28·5-s − 1.82·6-s − 1.96·7-s + 2.98·8-s − 0.0698·9-s + 2.42·10-s + 0.0323·11-s − 2.48·12-s − 3.72·14-s − 1.23·15-s + 3.06·16-s + 0.401·17-s − 0.132·18-s + 0.630·19-s + 3.30·20-s + 1.89·21-s + 0.0612·22-s − 0.427·23-s − 2.87·24-s + 0.642·25-s + 1.03·27-s − 5.07·28-s + 1.58·29-s − 2.33·30-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

\(L(1)\) \(\approx\) \(5.223495174\)
\(L(\frac12)\) \(\approx\) \(5.223495174\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.67T + 2T^{2} \)
3 \( 1 + 1.67T + 3T^{2} \)
5 \( 1 - 2.86T + 5T^{2} \)
7 \( 1 + 5.21T + 7T^{2} \)
11 \( 1 - 0.107T + 11T^{2} \)
17 \( 1 - 1.65T + 17T^{2} \)
19 \( 1 - 2.74T + 19T^{2} \)
23 \( 1 + 2.05T + 23T^{2} \)
29 \( 1 - 8.54T + 29T^{2} \)
37 \( 1 - 7.15T + 37T^{2} \)
41 \( 1 - 9.28T + 41T^{2} \)
43 \( 1 - 7.32T + 43T^{2} \)
47 \( 1 - 4.70T + 47T^{2} \)
53 \( 1 + 3.04T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 5.46T + 61T^{2} \)
67 \( 1 - 1.60T + 67T^{2} \)
71 \( 1 - 8.50T + 71T^{2} \)
73 \( 1 - 6.98T + 73T^{2} \)
79 \( 1 + 1.90T + 79T^{2} \)
83 \( 1 - 9.64T + 83T^{2} \)
89 \( 1 - 9.70T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−7.74699460806194879233946726565, −6.82838119170654600709540792504, −6.26831309098229731265635020092, −5.99607177027027930492252355817, −5.53164689828565401741453071696, −4.68844247262488856166047989809, −3.78381926246443651040759736802, −2.86532026903669651321483104450, −2.50041817609849460668581570945, −0.981314568525169403896780367462, 0.981314568525169403896780367462, 2.50041817609849460668581570945, 2.86532026903669651321483104450, 3.78381926246443651040759736802, 4.68844247262488856166047989809, 5.53164689828565401741453071696, 5.99607177027027930492252355817, 6.26831309098229731265635020092, 6.82838119170654600709540792504, 7.74699460806194879233946726565

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