Properties

Label 2-8015-1.1-c1-0-168
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.24·2-s + 2.46·3-s − 0.447·4-s − 5-s + 3.06·6-s − 7-s − 3.04·8-s + 3.06·9-s − 1.24·10-s + 1.46·11-s − 1.10·12-s − 4.42·13-s − 1.24·14-s − 2.46·15-s − 2.90·16-s − 0.0499·17-s + 3.81·18-s − 0.658·19-s + 0.447·20-s − 2.46·21-s + 1.82·22-s + 4.57·23-s − 7.50·24-s + 25-s − 5.50·26-s + 0.155·27-s + 0.447·28-s + ⋯
L(s)  = 1  + 0.881·2-s + 1.42·3-s − 0.223·4-s − 0.447·5-s + 1.25·6-s − 0.377·7-s − 1.07·8-s + 1.02·9-s − 0.394·10-s + 0.441·11-s − 0.317·12-s − 1.22·13-s − 0.333·14-s − 0.635·15-s − 0.726·16-s − 0.0121·17-s + 0.899·18-s − 0.151·19-s + 0.0999·20-s − 0.537·21-s + 0.388·22-s + 0.953·23-s − 1.53·24-s + 0.200·25-s − 1.08·26-s + 0.0299·27-s + 0.0845·28-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: \(0\)
Selberg data: \((2,\ 8015,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.659006324\)
\(L(\frac12)\) \(\approx\) \(3.659006324\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.24T + 2T^{2} \)
3 \( 1 - 2.46T + 3T^{2} \)
11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + 4.42T + 13T^{2} \)
17 \( 1 + 0.0499T + 17T^{2} \)
19 \( 1 + 0.658T + 19T^{2} \)
23 \( 1 - 4.57T + 23T^{2} \)
29 \( 1 - 4.16T + 29T^{2} \)
31 \( 1 + 5.17T + 31T^{2} \)
37 \( 1 - 2.60T + 37T^{2} \)
41 \( 1 - 6.25T + 41T^{2} \)
43 \( 1 - 5.47T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 8.42T + 53T^{2} \)
59 \( 1 - 8.86T + 59T^{2} \)
61 \( 1 - 9.68T + 61T^{2} \)
67 \( 1 + 1.18T + 67T^{2} \)
71 \( 1 + 4.70T + 71T^{2} \)
73 \( 1 - 8.22T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + 8.75T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 - 5.42T + 97T^{2} \)
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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

−7.76464057539674960095496435575, −7.24863235474512230162131128366, −6.51152747688855790025537088418, −5.54336441761227748299750740535, −4.86024558758017951425384952885, −3.97783263200322330528855861828, −3.72572341271404007914838510758, −2.68787259977514373641995657321, −2.42431564783367056840139450444, −0.75231907309159181460776107694, 0.75231907309159181460776107694, 2.42431564783367056840139450444, 2.68787259977514373641995657321, 3.72572341271404007914838510758, 3.97783263200322330528855861828, 4.86024558758017951425384952885, 5.54336441761227748299750740535, 6.51152747688855790025537088418, 7.24863235474512230162131128366, 7.76464057539674960095496435575

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