Properties

Label 2-8015-1.1-c1-0-439
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  + 1.84·2-s + 1.50·3-s + 1.40·4-s + 5-s + 2.78·6-s + 7-s − 1.10·8-s − 0.725·9-s + 1.84·10-s − 0.364·11-s + 2.11·12-s − 4.79·13-s + 1.84·14-s + 1.50·15-s − 4.83·16-s − 2.87·17-s − 1.33·18-s − 4.60·19-s + 1.40·20-s + 1.50·21-s − 0.671·22-s + 0.362·23-s − 1.66·24-s + 25-s − 8.85·26-s − 5.61·27-s + 1.40·28-s + ⋯
L(s)  = 1  + 1.30·2-s + 0.870·3-s + 0.700·4-s + 0.447·5-s + 1.13·6-s + 0.377·7-s − 0.390·8-s − 0.241·9-s + 0.583·10-s − 0.109·11-s + 0.610·12-s − 1.33·13-s + 0.492·14-s + 0.389·15-s − 1.20·16-s − 0.696·17-s − 0.315·18-s − 1.05·19-s + 0.313·20-s + 0.329·21-s − 0.143·22-s + 0.0755·23-s − 0.339·24-s + 0.200·25-s − 1.73·26-s − 1.08·27-s + 0.264·28-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 \( 1 - 1.84T + 2T^{2} \)
3 \( 1 - 1.50T + 3T^{2} \)
11 \( 1 + 0.364T + 11T^{2} \)
13 \( 1 + 4.79T + 13T^{2} \)
17 \( 1 + 2.87T + 17T^{2} \)
19 \( 1 + 4.60T + 19T^{2} \)
23 \( 1 - 0.362T + 23T^{2} \)
29 \( 1 + 1.18T + 29T^{2} \)
31 \( 1 - 6.71T + 31T^{2} \)
37 \( 1 - 7.09T + 37T^{2} \)
41 \( 1 + 0.0307T + 41T^{2} \)
43 \( 1 + 6.19T + 43T^{2} \)
47 \( 1 - 5.81T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 + 4.40T + 59T^{2} \)
61 \( 1 - 2.83T + 61T^{2} \)
67 \( 1 + 4.31T + 67T^{2} \)
71 \( 1 + 4.80T + 71T^{2} \)
73 \( 1 - 5.26T + 73T^{2} \)
79 \( 1 + 3.24T + 79T^{2} \)
83 \( 1 + 8.11T + 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + 9.86T + 97T^{2} \)
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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

−7.41598060453409341447758231383, −6.58614622768600461480138971539, −6.01036003268151823967465501144, −5.21614999989209080678484702120, −4.57187943647163885512521857750, −4.08246880080575801728316503713, −2.93712652417837388125750066131, −2.62021939524974728738034808384, −1.84556677137016756924834903139, 0, 1.84556677137016756924834903139, 2.62021939524974728738034808384, 2.93712652417837388125750066131, 4.08246880080575801728316503713, 4.57187943647163885512521857750, 5.21614999989209080678484702120, 6.01036003268151823967465501144, 6.58614622768600461480138971539, 7.41598060453409341447758231383

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