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  + 38.7·2-s − 81·3-s + 992.·4-s + 625·5-s − 3.14e3·6-s + 1.22e4·7-s + 1.86e4·8-s + 6.56e3·9-s + 2.42e4·10-s − 6.58e4·11-s − 8.03e4·12-s − 1.12e5·13-s + 4.75e5·14-s − 5.06e4·15-s + 2.14e5·16-s + 9.66e4·17-s + 2.54e5·18-s − 1.81e5·19-s + 6.20e5·20-s − 9.94e5·21-s − 2.55e6·22-s + 2.44e5·23-s − 1.50e6·24-s + 3.90e5·25-s − 4.35e6·26-s − 5.31e5·27-s + 1.21e7·28-s + ⋯
L(s)  = 1  + 1.71·2-s − 0.577·3-s + 1.93·4-s + 0.447·5-s − 0.989·6-s + 1.93·7-s + 1.60·8-s + 0.333·9-s + 0.766·10-s − 1.35·11-s − 1.11·12-s − 1.09·13-s + 3.31·14-s − 0.258·15-s + 0.818·16-s + 0.280·17-s + 0.571·18-s − 0.319·19-s + 0.866·20-s − 1.11·21-s − 2.32·22-s + 0.181·23-s − 0.928·24-s + 0.200·25-s − 1.86·26-s − 0.192·27-s + 3.74·28-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

\(L(5)\) \(\approx\) \(3.98079\)
\(L(\frac12)\) \(\approx\) \(3.98079\)
\(L(\frac{11}{2})\) 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 \( 1 - 38.7T + 512T^{2} \)
7 \( 1 - 1.22e4T + 4.03e7T^{2} \)
11 \( 1 + 6.58e4T + 2.35e9T^{2} \)
13 \( 1 + 1.12e5T + 1.06e10T^{2} \)
17 \( 1 - 9.66e4T + 1.18e11T^{2} \)
19 \( 1 + 1.81e5T + 3.22e11T^{2} \)
23 \( 1 - 2.44e5T + 1.80e12T^{2} \)
29 \( 1 + 5.26e6T + 1.45e13T^{2} \)
31 \( 1 - 1.83e6T + 2.64e13T^{2} \)
37 \( 1 - 6.36e6T + 1.29e14T^{2} \)
41 \( 1 - 1.57e6T + 3.27e14T^{2} \)
43 \( 1 + 1.99e7T + 5.02e14T^{2} \)
47 \( 1 - 3.00e7T + 1.11e15T^{2} \)
53 \( 1 + 2.57e6T + 3.29e15T^{2} \)
59 \( 1 + 1.19e8T + 8.66e15T^{2} \)
61 \( 1 - 1.92e8T + 1.16e16T^{2} \)
67 \( 1 + 1.20e8T + 2.72e16T^{2} \)
71 \( 1 + 6.99e5T + 4.58e16T^{2} \)
73 \( 1 + 8.91e7T + 5.88e16T^{2} \)
79 \( 1 - 4.31e8T + 1.19e17T^{2} \)
83 \( 1 + 1.69e7T + 1.86e17T^{2} \)
89 \( 1 + 3.09e6T + 3.50e17T^{2} \)
97 \( 1 - 5.72e8T + 7.60e17T^{2} \)
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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

−16.99940123368544952783131106395, −15.27174069726397164897883445595, −14.39850785303878044264511277679, −13.09087010916688159790936977715, −11.80263158849163533957508601208, −10.69818121115332193298987552172, −7.57020401659824680483034778008, −5.49769070495138489091464465718, −4.72368706125592097975057746691, −2.16767088918043568698981287728, 2.16767088918043568698981287728, 4.72368706125592097975057746691, 5.49769070495138489091464465718, 7.57020401659824680483034778008, 10.69818121115332193298987552172, 11.80263158849163533957508601208, 13.09087010916688159790936977715, 14.39850785303878044264511277679, 15.27174069726397164897883445595, 16.99940123368544952783131106395

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