Properties

Label 2-8015-1.1-c1-0-445
Degree $2$
Conductor $8015$
Sign $-1$
Analytic cond. $64.0000$
Root an. cond. $8.00000$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s + 7-s − 2·9-s + 2·10-s − 3·11-s + 2·12-s + 3·13-s + 2·14-s + 15-s − 4·16-s − 17-s − 4·18-s − 6·19-s + 2·20-s + 21-s − 6·22-s − 2·23-s + 25-s + 6·26-s − 5·27-s + 2·28-s − 3·29-s + 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 0.377·7-s − 2/3·9-s + 0.632·10-s − 0.904·11-s + 0.577·12-s + 0.832·13-s + 0.534·14-s + 0.258·15-s − 16-s − 0.242·17-s − 0.942·18-s − 1.37·19-s + 0.447·20-s + 0.218·21-s − 1.27·22-s − 0.417·23-s + 1/5·25-s + 1.17·26-s − 0.962·27-s + 0.377·28-s − 0.557·29-s + 0.365·30-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: yes
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 \( 1 - p T + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−7.40609303321488918737352136002, −6.39729820283455888127193725754, −6.05547334061216319470775397163, −5.23644784185415968431036583531, −4.77851124132374451098118288248, −3.77656060833449355859079744944, −3.32873751151844003369048756236, −2.36360758653652029637163980202, −1.90024837111624115231112726687, 0, 1.90024837111624115231112726687, 2.36360758653652029637163980202, 3.32873751151844003369048756236, 3.77656060833449355859079744944, 4.77851124132374451098118288248, 5.23644784185415968431036583531, 6.05547334061216319470775397163, 6.39729820283455888127193725754, 7.40609303321488918737352136002

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