Properties

Label 2-8015-1.1-c1-0-359
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  − 0.139·2-s + 2.19·3-s − 1.98·4-s − 5-s − 0.307·6-s − 7-s + 0.557·8-s + 1.82·9-s + 0.139·10-s + 0.610·11-s − 4.34·12-s + 0.265·13-s + 0.139·14-s − 2.19·15-s + 3.88·16-s + 3.41·17-s − 0.254·18-s + 1.47·19-s + 1.98·20-s − 2.19·21-s − 0.0854·22-s − 4.58·23-s + 1.22·24-s + 25-s − 0.0370·26-s − 2.58·27-s + 1.98·28-s + ⋯
L(s)  = 1  − 0.0989·2-s + 1.26·3-s − 0.990·4-s − 0.447·5-s − 0.125·6-s − 0.377·7-s + 0.196·8-s + 0.607·9-s + 0.0442·10-s + 0.184·11-s − 1.25·12-s + 0.0735·13-s + 0.0373·14-s − 0.566·15-s + 0.970·16-s + 0.827·17-s − 0.0600·18-s + 0.337·19-s + 0.442·20-s − 0.479·21-s − 0.0182·22-s − 0.957·23-s + 0.249·24-s + 0.200·25-s − 0.00727·26-s − 0.497·27-s + 0.374·28-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 \( 1 + 0.139T + 2T^{2} \)
3 \( 1 - 2.19T + 3T^{2} \)
11 \( 1 - 0.610T + 11T^{2} \)
13 \( 1 - 0.265T + 13T^{2} \)
17 \( 1 - 3.41T + 17T^{2} \)
19 \( 1 - 1.47T + 19T^{2} \)
23 \( 1 + 4.58T + 23T^{2} \)
29 \( 1 + 5.76T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 + 0.632T + 41T^{2} \)
43 \( 1 - 5.13T + 43T^{2} \)
47 \( 1 - 6.73T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 7.58T + 59T^{2} \)
61 \( 1 + 5.74T + 61T^{2} \)
67 \( 1 - 3.66T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 7.02T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 5.87T + 83T^{2} \)
89 \( 1 + 8.68T + 89T^{2} \)
97 \( 1 - 3.34T + 97T^{2} \)
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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

−7.60955567740035985326654563788, −7.25710779371820747425406110570, −5.86954480368703904135543102322, −5.51165794879017600043613642759, −4.25798022665766096952022515613, −3.87517341153088301595288122288, −3.26599739893234469976574526908, −2.38086176916271823233539648606, −1.26299638856228772728393892859, 0, 1.26299638856228772728393892859, 2.38086176916271823233539648606, 3.26599739893234469976574526908, 3.87517341153088301595288122288, 4.25798022665766096952022515613, 5.51165794879017600043613642759, 5.86954480368703904135543102322, 7.25710779371820747425406110570, 7.60955567740035985326654563788

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