Properties

Label 2-8015-1.1-c1-0-312
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  − 0.978·2-s − 3.27·3-s − 1.04·4-s − 5-s + 3.20·6-s − 7-s + 2.97·8-s + 7.74·9-s + 0.978·10-s + 6.25·11-s + 3.41·12-s + 0.406·13-s + 0.978·14-s + 3.27·15-s − 0.824·16-s + 6.95·17-s − 7.57·18-s + 0.159·19-s + 1.04·20-s + 3.27·21-s − 6.11·22-s + 3.08·23-s − 9.75·24-s + 25-s − 0.397·26-s − 15.5·27-s + 1.04·28-s + ⋯
L(s)  = 1  − 0.691·2-s − 1.89·3-s − 0.521·4-s − 0.447·5-s + 1.30·6-s − 0.377·7-s + 1.05·8-s + 2.58·9-s + 0.309·10-s + 1.88·11-s + 0.987·12-s + 0.112·13-s + 0.261·14-s + 0.846·15-s − 0.206·16-s + 1.68·17-s − 1.78·18-s + 0.0365·19-s + 0.233·20-s + 0.715·21-s − 1.30·22-s + 0.644·23-s − 1.99·24-s + 0.200·25-s − 0.0779·26-s − 2.99·27-s + 0.197·28-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 \( 1 + 0.978T + 2T^{2} \)
3 \( 1 + 3.27T + 3T^{2} \)
11 \( 1 - 6.25T + 11T^{2} \)
13 \( 1 - 0.406T + 13T^{2} \)
17 \( 1 - 6.95T + 17T^{2} \)
19 \( 1 - 0.159T + 19T^{2} \)
23 \( 1 - 3.08T + 23T^{2} \)
29 \( 1 - 2.86T + 29T^{2} \)
31 \( 1 + 2.48T + 31T^{2} \)
37 \( 1 + 0.474T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + 9.49T + 47T^{2} \)
53 \( 1 + 2.93T + 53T^{2} \)
59 \( 1 + 0.928T + 59T^{2} \)
61 \( 1 + 0.713T + 61T^{2} \)
67 \( 1 + 4.27T + 67T^{2} \)
71 \( 1 - 4.68T + 71T^{2} \)
73 \( 1 + 7.43T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + 7.78T + 83T^{2} \)
89 \( 1 - 9.37T + 89T^{2} \)
97 \( 1 + 8.66T + 97T^{2} \)
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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

−7.34501286801389843320440235855, −6.77533589077562918196920821887, −6.19178530059376491735551176010, −5.39858268526693721335745224668, −4.79967031451653126972324228973, −4.02586228816774473143885032306, −3.47619487176962939160763106049, −1.38071603572512385727191989718, −1.08529431653670817187799754898, 0, 1.08529431653670817187799754898, 1.38071603572512385727191989718, 3.47619487176962939160763106049, 4.02586228816774473143885032306, 4.79967031451653126972324228973, 5.39858268526693721335745224668, 6.19178530059376491735551176010, 6.77533589077562918196920821887, 7.34501286801389843320440235855

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