Properties

Label 2-8015-1.1-c1-0-310
Degree $2$
Conductor $8015$
Sign $-1$
Analytic cond. $64.0000$
Root an. cond. $8.00000$
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  − 1.65·2-s + 1.21·3-s + 0.751·4-s + 5-s − 2.01·6-s − 7-s + 2.07·8-s − 1.52·9-s − 1.65·10-s − 5.02·11-s + 0.912·12-s − 1.70·13-s + 1.65·14-s + 1.21·15-s − 4.93·16-s + 4.99·17-s + 2.52·18-s − 0.872·19-s + 0.751·20-s − 1.21·21-s + 8.33·22-s − 2.73·23-s + 2.51·24-s + 25-s + 2.81·26-s − 5.49·27-s − 0.751·28-s + ⋯
L(s)  = 1  − 1.17·2-s + 0.701·3-s + 0.375·4-s + 0.447·5-s − 0.822·6-s − 0.377·7-s + 0.732·8-s − 0.507·9-s − 0.524·10-s − 1.51·11-s + 0.263·12-s − 0.471·13-s + 0.443·14-s + 0.313·15-s − 1.23·16-s + 1.21·17-s + 0.595·18-s − 0.200·19-s + 0.167·20-s − 0.265·21-s + 1.77·22-s − 0.569·23-s + 0.513·24-s + 0.200·25-s + 0.553·26-s − 1.05·27-s − 0.141·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8015\)    =    \(5 \cdot 7 \cdot 229\)
Sign: $-1$
Analytic conductor: \(64.0000\)
Root analytic conductor: \(8.00000\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8015,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
7 \( 1 + T \)
229 \( 1 - T \)
good2 \( 1 + 1.65T + 2T^{2} \)
3 \( 1 - 1.21T + 3T^{2} \)
11 \( 1 + 5.02T + 11T^{2} \)
13 \( 1 + 1.70T + 13T^{2} \)
17 \( 1 - 4.99T + 17T^{2} \)
19 \( 1 + 0.872T + 19T^{2} \)
23 \( 1 + 2.73T + 23T^{2} \)
29 \( 1 - 0.150T + 29T^{2} \)
31 \( 1 - 7.20T + 31T^{2} \)
37 \( 1 - 4.43T + 37T^{2} \)
41 \( 1 - 6.95T + 41T^{2} \)
43 \( 1 - 3.64T + 43T^{2} \)
47 \( 1 + 4.69T + 47T^{2} \)
53 \( 1 - 5.70T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 + 0.319T + 61T^{2} \)
67 \( 1 - 1.80T + 67T^{2} \)
71 \( 1 + 3.63T + 71T^{2} \)
73 \( 1 + 6.37T + 73T^{2} \)
79 \( 1 - 6.92T + 79T^{2} \)
83 \( 1 + 2.79T + 83T^{2} \)
89 \( 1 + 7.82T + 89T^{2} \)
97 \( 1 + 7.30T + 97T^{2} \)
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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

−7.75894742052532462847818801210, −7.21984447539501606371224248326, −6.13850753257703605150584811149, −5.49893761284264970290416526787, −4.73027265016814496009075205962, −3.71776955690330762985200079631, −2.61072609497594210279070109382, −2.41870768061837310643501278439, −1.07799870319071291645807112478, 0, 1.07799870319071291645807112478, 2.41870768061837310643501278439, 2.61072609497594210279070109382, 3.71776955690330762985200079631, 4.73027265016814496009075205962, 5.49893761284264970290416526787, 6.13850753257703605150584811149, 7.21984447539501606371224248326, 7.75894742052532462847818801210

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