Properties

Label 2-8015-1.1-c1-0-262
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  − 1.65·2-s − 1.94·3-s + 0.754·4-s − 5-s + 3.22·6-s + 7-s + 2.06·8-s + 0.767·9-s + 1.65·10-s − 1.86·11-s − 1.46·12-s + 5.05·13-s − 1.65·14-s + 1.94·15-s − 4.93·16-s + 5.41·17-s − 1.27·18-s − 0.656·19-s − 0.754·20-s − 1.94·21-s + 3.09·22-s + 1.52·23-s − 4.01·24-s + 25-s − 8.38·26-s + 4.33·27-s + 0.754·28-s + ⋯
L(s)  = 1  − 1.17·2-s − 1.12·3-s + 0.377·4-s − 0.447·5-s + 1.31·6-s + 0.377·7-s + 0.730·8-s + 0.255·9-s + 0.524·10-s − 0.563·11-s − 0.422·12-s + 1.40·13-s − 0.443·14-s + 0.501·15-s − 1.23·16-s + 1.31·17-s − 0.300·18-s − 0.150·19-s − 0.168·20-s − 0.423·21-s + 0.660·22-s + 0.317·23-s − 0.819·24-s + 0.200·25-s − 1.64·26-s + 0.833·27-s + 0.142·28-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 \( 1 + 1.65T + 2T^{2} \)
3 \( 1 + 1.94T + 3T^{2} \)
11 \( 1 + 1.86T + 11T^{2} \)
13 \( 1 - 5.05T + 13T^{2} \)
17 \( 1 - 5.41T + 17T^{2} \)
19 \( 1 + 0.656T + 19T^{2} \)
23 \( 1 - 1.52T + 23T^{2} \)
29 \( 1 - 2.97T + 29T^{2} \)
31 \( 1 + 0.866T + 31T^{2} \)
37 \( 1 + 5.13T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 + 7.99T + 43T^{2} \)
47 \( 1 + 8.07T + 47T^{2} \)
53 \( 1 - 3.90T + 53T^{2} \)
59 \( 1 + 9.44T + 59T^{2} \)
61 \( 1 + 6.35T + 61T^{2} \)
67 \( 1 - 8.23T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 5.83T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 3.77T + 83T^{2} \)
89 \( 1 + 3.71T + 89T^{2} \)
97 \( 1 - 8.86T + 97T^{2} \)
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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

−7.75636078223576973473413663823, −6.79170504501392662395565244245, −6.31712518256733044910545384456, −5.21144999104295693072816639464, −5.05663687271232037756003659223, −3.91677204878791519728905997398, −3.10497836776422249291794415366, −1.66139100154074774566185745688, −0.962553622715792141076987139273, 0, 0.962553622715792141076987139273, 1.66139100154074774566185745688, 3.10497836776422249291794415366, 3.91677204878791519728905997398, 5.05663687271232037756003659223, 5.21144999104295693072816639464, 6.31712518256733044910545384456, 6.79170504501392662395565244245, 7.75636078223576973473413663823

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