Properties

Degree $2$
Conductor $15$
Sign $1$
Motivic weight $9$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.78·2-s − 81·3-s − 451.·4-s + 625·5-s + 630.·6-s + 1.83e3·7-s + 7.50e3·8-s + 6.56e3·9-s − 4.86e3·10-s + 4.43e4·11-s + 3.65e4·12-s + 1.36e5·13-s − 1.43e4·14-s − 5.06e4·15-s + 1.72e5·16-s − 2.53e5·17-s − 5.10e4·18-s + 8.54e4·19-s − 2.82e5·20-s − 1.49e5·21-s − 3.45e5·22-s − 9.79e5·23-s − 6.07e5·24-s + 3.90e5·25-s − 1.06e6·26-s − 5.31e5·27-s − 8.30e5·28-s + ⋯
L(s)  = 1  − 0.344·2-s − 0.577·3-s − 0.881·4-s + 0.447·5-s + 0.198·6-s + 0.289·7-s + 0.647·8-s + 0.333·9-s − 0.153·10-s + 0.914·11-s + 0.508·12-s + 1.32·13-s − 0.0996·14-s − 0.258·15-s + 0.658·16-s − 0.736·17-s − 0.114·18-s + 0.150·19-s − 0.394·20-s − 0.167·21-s − 0.314·22-s − 0.729·23-s − 0.373·24-s + 0.200·25-s − 0.456·26-s − 0.192·27-s − 0.255·28-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: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.11408\)
\(L(\frac12)\) \(\approx\) \(1.11408\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81T \)
5 \( 1 - 625T \)
good2 \( 1 + 7.78T + 512T^{2} \)
7 \( 1 - 1.83e3T + 4.03e7T^{2} \)
11 \( 1 - 4.43e4T + 2.35e9T^{2} \)
13 \( 1 - 1.36e5T + 1.06e10T^{2} \)
17 \( 1 + 2.53e5T + 1.18e11T^{2} \)
19 \( 1 - 8.54e4T + 3.22e11T^{2} \)
23 \( 1 + 9.79e5T + 1.80e12T^{2} \)
29 \( 1 - 2.58e6T + 1.45e13T^{2} \)
31 \( 1 - 8.94e6T + 2.64e13T^{2} \)
37 \( 1 - 1.56e7T + 1.29e14T^{2} \)
41 \( 1 - 2.44e7T + 3.27e14T^{2} \)
43 \( 1 - 1.27e7T + 5.02e14T^{2} \)
47 \( 1 + 6.16e7T + 1.11e15T^{2} \)
53 \( 1 - 5.70e6T + 3.29e15T^{2} \)
59 \( 1 - 8.35e7T + 8.66e15T^{2} \)
61 \( 1 - 1.48e8T + 1.16e16T^{2} \)
67 \( 1 + 1.68e8T + 2.72e16T^{2} \)
71 \( 1 - 2.10e8T + 4.58e16T^{2} \)
73 \( 1 + 1.43e8T + 5.88e16T^{2} \)
79 \( 1 + 4.55e8T + 1.19e17T^{2} \)
83 \( 1 + 3.55e8T + 1.86e17T^{2} \)
89 \( 1 + 4.24e8T + 3.50e17T^{2} \)
97 \( 1 - 1.19e9T + 7.60e17T^{2} \)
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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

−17.50879103791912782747065835903, −16.14276913180374231901880793526, −14.25474033065510341558678295690, −13.11268265924733439960737401695, −11.36294620416704591200972006337, −9.834405788870403866626550613586, −8.439094873599952929310718602775, −6.20036582165188861355970623800, −4.35095565650486616548205522556, −1.08218705022902846707049609956, 1.08218705022902846707049609956, 4.35095565650486616548205522556, 6.20036582165188861355970623800, 8.439094873599952929310718602775, 9.834405788870403866626550613586, 11.36294620416704591200972006337, 13.11268265924733439960737401695, 14.25474033065510341558678295690, 16.14276913180374231901880793526, 17.50879103791912782747065835903

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