Properties

Label 2-8015-1.1-c1-0-363
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  − 2.58·2-s + 1.46·3-s + 4.69·4-s − 5-s − 3.80·6-s + 7-s − 6.98·8-s − 0.844·9-s + 2.58·10-s + 3.30·11-s + 6.89·12-s + 1.30·13-s − 2.58·14-s − 1.46·15-s + 8.68·16-s + 6.35·17-s + 2.18·18-s − 2.58·19-s − 4.69·20-s + 1.46·21-s − 8.56·22-s − 5.17·23-s − 10.2·24-s + 25-s − 3.38·26-s − 5.64·27-s + 4.69·28-s + ⋯
L(s)  = 1  − 1.83·2-s + 0.847·3-s + 2.34·4-s − 0.447·5-s − 1.55·6-s + 0.377·7-s − 2.46·8-s − 0.281·9-s + 0.818·10-s + 0.997·11-s + 1.99·12-s + 0.363·13-s − 0.691·14-s − 0.379·15-s + 2.17·16-s + 1.54·17-s + 0.515·18-s − 0.591·19-s − 1.05·20-s + 0.320·21-s − 1.82·22-s − 1.07·23-s − 2.09·24-s + 0.200·25-s − 0.664·26-s − 1.08·27-s + 0.887·28-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 \( 1 + 2.58T + 2T^{2} \)
3 \( 1 - 1.46T + 3T^{2} \)
11 \( 1 - 3.30T + 11T^{2} \)
13 \( 1 - 1.30T + 13T^{2} \)
17 \( 1 - 6.35T + 17T^{2} \)
19 \( 1 + 2.58T + 19T^{2} \)
23 \( 1 + 5.17T + 23T^{2} \)
29 \( 1 - 3.24T + 29T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 - 6.54T + 37T^{2} \)
41 \( 1 + 1.04T + 41T^{2} \)
43 \( 1 + 9.04T + 43T^{2} \)
47 \( 1 + 2.12T + 47T^{2} \)
53 \( 1 + 7.79T + 53T^{2} \)
59 \( 1 + 7.35T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 7.75T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 6.30T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 - 6.92T + 89T^{2} \)
97 \( 1 + 9.19T + 97T^{2} \)
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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

−7.80262917040089983425215391929, −7.25615069176191832584324471766, −6.32116333402104235934759332563, −5.84178683189170299351777915107, −4.48218387190423437410792693218, −3.49502178495072178560006467160, −2.93432667315129174431088923421, −1.85321776246029950949876319302, −1.27899758500321722124702445155, 0, 1.27899758500321722124702445155, 1.85321776246029950949876319302, 2.93432667315129174431088923421, 3.49502178495072178560006467160, 4.48218387190423437410792693218, 5.84178683189170299351777915107, 6.32116333402104235934759332563, 7.25615069176191832584324471766, 7.80262917040089983425215391929

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