Properties

Degree 2
Conductor $ 3 \cdot 5 $
Sign $1$
Motivic weight 11
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.33·2-s + 243·3-s − 2.04e3·4-s + 3.12e3·5-s + 323.·6-s + 3.94e4·7-s − 5.44e3·8-s + 5.90e4·9-s + 4.15e3·10-s − 1.37e5·11-s − 4.97e5·12-s + 2.54e6·13-s + 5.25e4·14-s + 7.59e5·15-s + 4.18e6·16-s + 1.02e7·17-s + 7.85e4·18-s − 1.05e7·19-s − 6.39e6·20-s + 9.59e6·21-s − 1.82e5·22-s + 2.32e7·23-s − 1.32e6·24-s + 9.76e6·25-s + 3.39e6·26-s + 1.43e7·27-s − 8.07e7·28-s + ⋯
L(s)  = 1  + 0.0294·2-s + 0.577·3-s − 0.999·4-s + 0.447·5-s + 0.0169·6-s + 0.887·7-s − 0.0587·8-s + 0.333·9-s + 0.0131·10-s − 0.256·11-s − 0.576·12-s + 1.90·13-s + 0.0261·14-s + 0.258·15-s + 0.997·16-s + 1.75·17-s + 0.00980·18-s − 0.973·19-s − 0.446·20-s + 0.512·21-s − 0.00755·22-s + 0.752·23-s − 0.0339·24-s + 0.199·25-s + 0.0560·26-s + 0.192·27-s − 0.887·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  $\chi_{15} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 15,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $2.17421$
$L(\frac12)$  $\approx$  $2.17421$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - 243T \)
5 \( 1 - 3.12e3T \)
good2 \( 1 - 1.33T + 2.04e3T^{2} \)
7 \( 1 - 3.94e4T + 1.97e9T^{2} \)
11 \( 1 + 1.37e5T + 2.85e11T^{2} \)
13 \( 1 - 2.54e6T + 1.79e12T^{2} \)
17 \( 1 - 1.02e7T + 3.42e13T^{2} \)
19 \( 1 + 1.05e7T + 1.16e14T^{2} \)
23 \( 1 - 2.32e7T + 9.52e14T^{2} \)
29 \( 1 + 1.59e8T + 1.22e16T^{2} \)
31 \( 1 + 1.39e8T + 2.54e16T^{2} \)
37 \( 1 + 2.50e8T + 1.77e17T^{2} \)
41 \( 1 - 8.03e8T + 5.50e17T^{2} \)
43 \( 1 + 6.15e8T + 9.29e17T^{2} \)
47 \( 1 - 1.40e9T + 2.47e18T^{2} \)
53 \( 1 - 2.24e9T + 9.26e18T^{2} \)
59 \( 1 + 5.57e9T + 3.01e19T^{2} \)
61 \( 1 - 2.46e9T + 4.35e19T^{2} \)
67 \( 1 + 9.05e9T + 1.22e20T^{2} \)
71 \( 1 - 2.45e10T + 2.31e20T^{2} \)
73 \( 1 - 2.54e9T + 3.13e20T^{2} \)
79 \( 1 - 1.79e10T + 7.47e20T^{2} \)
83 \( 1 - 1.40e10T + 1.28e21T^{2} \)
89 \( 1 + 6.35e10T + 2.77e21T^{2} \)
97 \( 1 + 8.52e10T + 7.15e21T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.79420695935297079114706960108, −14.91285971819391167289483854206, −13.91922694574203468931879600809, −12.85982373338608671803792737886, −10.74654215327138089288277065645, −9.122848298850613298242115854820, −8.025314805281953920685288232571, −5.51790093004423822430495691707, −3.72981044516741783007100280936, −1.33084012987036030831987407700, 1.33084012987036030831987407700, 3.72981044516741783007100280936, 5.51790093004423822430495691707, 8.025314805281953920685288232571, 9.122848298850613298242115854820, 10.74654215327138089288277065645, 12.85982373338608671803792737886, 13.91922694574203468931879600809, 14.91285971819391167289483854206, 16.79420695935297079114706960108

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