Properties

Label 2-8015-1.1-c1-0-145
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  − 1.67·2-s − 1.48·3-s + 0.810·4-s − 5-s + 2.49·6-s + 7-s + 1.99·8-s − 0.792·9-s + 1.67·10-s − 3.82·11-s − 1.20·12-s − 6.09·13-s − 1.67·14-s + 1.48·15-s − 4.96·16-s − 2.99·17-s + 1.32·18-s − 3.51·19-s − 0.810·20-s − 1.48·21-s + 6.41·22-s − 3.88·23-s − 2.96·24-s + 25-s + 10.2·26-s + 5.63·27-s + 0.810·28-s + ⋯
L(s)  = 1  − 1.18·2-s − 0.857·3-s + 0.405·4-s − 0.447·5-s + 1.01·6-s + 0.377·7-s + 0.704·8-s − 0.264·9-s + 0.530·10-s − 1.15·11-s − 0.347·12-s − 1.68·13-s − 0.448·14-s + 0.383·15-s − 1.24·16-s − 0.727·17-s + 0.313·18-s − 0.807·19-s − 0.181·20-s − 0.324·21-s + 1.36·22-s − 0.809·23-s − 0.604·24-s + 0.200·25-s + 2.00·26-s + 1.08·27-s + 0.153·28-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 \( 1 + 1.67T + 2T^{2} \)
3 \( 1 + 1.48T + 3T^{2} \)
11 \( 1 + 3.82T + 11T^{2} \)
13 \( 1 + 6.09T + 13T^{2} \)
17 \( 1 + 2.99T + 17T^{2} \)
19 \( 1 + 3.51T + 19T^{2} \)
23 \( 1 + 3.88T + 23T^{2} \)
29 \( 1 - 6.72T + 29T^{2} \)
31 \( 1 - 2.22T + 31T^{2} \)
37 \( 1 + 8.02T + 37T^{2} \)
41 \( 1 - 5.71T + 41T^{2} \)
43 \( 1 - 9.86T + 43T^{2} \)
47 \( 1 - 5.24T + 47T^{2} \)
53 \( 1 - 3.02T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 - 7.95T + 61T^{2} \)
67 \( 1 - 5.74T + 67T^{2} \)
71 \( 1 + 2.42T + 71T^{2} \)
73 \( 1 - 9.64T + 73T^{2} \)
79 \( 1 + 5.43T + 79T^{2} \)
83 \( 1 - 9.76T + 83T^{2} \)
89 \( 1 + 4.12T + 89T^{2} \)
97 \( 1 + 1.99T + 97T^{2} \)
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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

−7.58943040263391276707783770302, −7.05539510273216607231147150918, −6.24356910021176522858530573372, −5.30503022774064884523135043941, −4.76603818906696832815097252514, −4.20836006774541995140162482554, −2.66773871806240046733875652310, −2.15641019031374933551244836833, −0.68907504271414041266448592329, 0, 0.68907504271414041266448592329, 2.15641019031374933551244836833, 2.66773871806240046733875652310, 4.20836006774541995140162482554, 4.76603818906696832815097252514, 5.30503022774064884523135043941, 6.24356910021176522858530573372, 7.05539510273216607231147150918, 7.58943040263391276707783770302

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