Properties

Degree $2$
Conductor $15$
Sign $-1$
Motivic weight $9$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 22·2-s − 81·3-s − 28·4-s − 625·5-s − 1.78e3·6-s − 5.98e3·7-s − 1.18e4·8-s + 6.56e3·9-s − 1.37e4·10-s − 1.46e4·11-s + 2.26e3·12-s + 3.79e4·13-s − 1.31e5·14-s + 5.06e4·15-s − 2.47e5·16-s − 4.41e5·17-s + 1.44e5·18-s + 4.41e5·19-s + 1.75e4·20-s + 4.85e5·21-s − 3.22e5·22-s + 2.26e6·23-s + 9.62e5·24-s + 3.90e5·25-s + 8.33e5·26-s − 5.31e5·27-s + 1.67e5·28-s + ⋯
L(s)  = 1  + 0.972·2-s − 0.577·3-s − 0.0546·4-s − 0.447·5-s − 0.561·6-s − 0.942·7-s − 1.02·8-s + 1/3·9-s − 0.434·10-s − 0.301·11-s + 0.0315·12-s + 0.368·13-s − 0.916·14-s + 0.258·15-s − 0.942·16-s − 1.28·17-s + 0.324·18-s + 0.777·19-s + 0.0244·20-s + 0.544·21-s − 0.293·22-s + 1.68·23-s + 0.592·24-s + 1/5·25-s + 0.357·26-s − 0.192·27-s + 0.0515·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-1$
Motivic weight: \(9\)
Character: $\chi_{15} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{4} T \)
5 \( 1 + p^{4} T \)
good2 \( 1 - 11 p T + p^{9} T^{2} \)
7 \( 1 + 5988 T + p^{9} T^{2} \)
11 \( 1 + 14648 T + p^{9} T^{2} \)
13 \( 1 - 37906 T + p^{9} T^{2} \)
17 \( 1 + 441098 T + p^{9} T^{2} \)
19 \( 1 - 441820 T + p^{9} T^{2} \)
23 \( 1 - 2264136 T + p^{9} T^{2} \)
29 \( 1 + 1049350 T + p^{9} T^{2} \)
31 \( 1 + 7910568 T + p^{9} T^{2} \)
37 \( 1 + 20992558 T + p^{9} T^{2} \)
41 \( 1 - 13285562 T + p^{9} T^{2} \)
43 \( 1 + 23130764 T + p^{9} T^{2} \)
47 \( 1 + 13873688 T + p^{9} T^{2} \)
53 \( 1 + 57635174 T + p^{9} T^{2} \)
59 \( 1 + 32042120 T + p^{9} T^{2} \)
61 \( 1 - 110664022 T + p^{9} T^{2} \)
67 \( 1 + 118568268 T + p^{9} T^{2} \)
71 \( 1 - 276679712 T + p^{9} T^{2} \)
73 \( 1 + 264023294 T + p^{9} T^{2} \)
79 \( 1 - 448202760 T + p^{9} T^{2} \)
83 \( 1 - 851015796 T + p^{9} T^{2} \)
89 \( 1 - 189894930 T + p^{9} T^{2} \)
97 \( 1 + 1014149278 T + p^{9} T^{2} \)
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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

−16.19639337037968540755731180359, −15.15456878796812744252248709069, −13.44082422659810418891079031245, −12.57749847910174203327583792004, −11.11356533531879127535454372456, −9.159381018473888421064584261067, −6.73698385780654091469689859497, −5.12695517312718126379745890763, −3.44606926321203345768029167895, 0, 3.44606926321203345768029167895, 5.12695517312718126379745890763, 6.73698385780654091469689859497, 9.159381018473888421064584261067, 11.11356533531879127535454372456, 12.57749847910174203327583792004, 13.44082422659810418891079031245, 15.15456878796812744252248709069, 16.19639337037968540755731180359

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