Properties

Label 2-8015-1.1-c1-0-281
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  − 0.525·2-s + 0.0103·3-s − 1.72·4-s − 5-s − 0.00545·6-s + 7-s + 1.95·8-s − 2.99·9-s + 0.525·10-s + 2.45·11-s − 0.0178·12-s − 2.86·13-s − 0.525·14-s − 0.0103·15-s + 2.41·16-s − 1.89·17-s + 1.57·18-s + 6.96·19-s + 1.72·20-s + 0.0103·21-s − 1.29·22-s − 1.83·23-s + 0.0203·24-s + 25-s + 1.50·26-s − 0.0622·27-s − 1.72·28-s + ⋯
L(s)  = 1  − 0.371·2-s + 0.00599·3-s − 0.861·4-s − 0.447·5-s − 0.00222·6-s + 0.377·7-s + 0.691·8-s − 0.999·9-s + 0.166·10-s + 0.740·11-s − 0.00516·12-s − 0.793·13-s − 0.140·14-s − 0.00268·15-s + 0.604·16-s − 0.458·17-s + 0.371·18-s + 1.59·19-s + 0.385·20-s + 0.00226·21-s − 0.275·22-s − 0.382·23-s + 0.00414·24-s + 0.200·25-s + 0.294·26-s − 0.0119·27-s − 0.325·28-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 \( 1 + 0.525T + 2T^{2} \)
3 \( 1 - 0.0103T + 3T^{2} \)
11 \( 1 - 2.45T + 11T^{2} \)
13 \( 1 + 2.86T + 13T^{2} \)
17 \( 1 + 1.89T + 17T^{2} \)
19 \( 1 - 6.96T + 19T^{2} \)
23 \( 1 + 1.83T + 23T^{2} \)
29 \( 1 + 8.45T + 29T^{2} \)
31 \( 1 - 3.05T + 31T^{2} \)
37 \( 1 + 6.07T + 37T^{2} \)
41 \( 1 - 9.43T + 41T^{2} \)
43 \( 1 + 0.608T + 43T^{2} \)
47 \( 1 + 3.36T + 47T^{2} \)
53 \( 1 - 5.23T + 53T^{2} \)
59 \( 1 + 3.40T + 59T^{2} \)
61 \( 1 - 1.78T + 61T^{2} \)
67 \( 1 + 4.69T + 67T^{2} \)
71 \( 1 - 16.6T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 - 2.47T + 83T^{2} \)
89 \( 1 + 4.96T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−7.63568488915483664921898410356, −7.06010232360531328453321633686, −5.96801096216017282396311832432, −5.31706080574494915042207363407, −4.71546263717416519483660560906, −3.88784062473758588808443068923, −3.24945998497930772997575595894, −2.14153108605284536351932808715, −1.01462239806734352963575357624, 0, 1.01462239806734352963575357624, 2.14153108605284536351932808715, 3.24945998497930772997575595894, 3.88784062473758588808443068923, 4.71546263717416519483660560906, 5.31706080574494915042207363407, 5.96801096216017282396311832432, 7.06010232360531328453321633686, 7.63568488915483664921898410356

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