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  − 74.9·2-s + 243·3-s + 3.57e3·4-s + 3.12e3·5-s − 1.82e4·6-s − 6.30e4·7-s − 1.14e5·8-s + 5.90e4·9-s − 2.34e5·10-s + 2.56e5·11-s + 8.68e5·12-s − 4.65e5·13-s + 4.72e6·14-s + 7.59e5·15-s + 1.25e6·16-s + 1.08e7·17-s − 4.42e6·18-s + 9.38e6·19-s + 1.11e7·20-s − 1.53e7·21-s − 1.92e7·22-s − 2.41e7·23-s − 2.77e7·24-s + 9.76e6·25-s + 3.49e7·26-s + 1.43e7·27-s − 2.25e8·28-s + ⋯
L(s)  = 1  − 1.65·2-s + 0.577·3-s + 1.74·4-s + 0.447·5-s − 0.956·6-s − 1.41·7-s − 1.23·8-s + 0.333·9-s − 0.740·10-s + 0.480·11-s + 1.00·12-s − 0.347·13-s + 2.34·14-s + 0.258·15-s + 0.299·16-s + 1.85·17-s − 0.552·18-s + 0.869·19-s + 0.780·20-s − 0.818·21-s − 0.795·22-s − 0.781·23-s − 0.712·24-s + 0.199·25-s + 0.576·26-s + 0.192·27-s − 2.47·28-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  =  $(2,\ 15,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $0.946166$
$L(\frac12)$  $\approx$  $0.946166$
$L(\frac{13}{2})$   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 \( 1 + 74.9T + 2.04e3T^{2} \)
7 \( 1 + 6.30e4T + 1.97e9T^{2} \)
11 \( 1 - 2.56e5T + 2.85e11T^{2} \)
13 \( 1 + 4.65e5T + 1.79e12T^{2} \)
17 \( 1 - 1.08e7T + 3.42e13T^{2} \)
19 \( 1 - 9.38e6T + 1.16e14T^{2} \)
23 \( 1 + 2.41e7T + 9.52e14T^{2} \)
29 \( 1 - 2.07e8T + 1.22e16T^{2} \)
31 \( 1 - 2.06e8T + 2.54e16T^{2} \)
37 \( 1 - 2.18e8T + 1.77e17T^{2} \)
41 \( 1 - 8.54e8T + 5.50e17T^{2} \)
43 \( 1 + 2.70e8T + 9.29e17T^{2} \)
47 \( 1 + 2.09e9T + 2.47e18T^{2} \)
53 \( 1 + 1.20e9T + 9.26e18T^{2} \)
59 \( 1 - 6.04e9T + 3.01e19T^{2} \)
61 \( 1 + 1.28e10T + 4.35e19T^{2} \)
67 \( 1 - 6.01e9T + 1.22e20T^{2} \)
71 \( 1 - 7.99e9T + 2.31e20T^{2} \)
73 \( 1 - 3.23e10T + 3.13e20T^{2} \)
79 \( 1 - 3.84e9T + 7.47e20T^{2} \)
83 \( 1 + 1.29e9T + 1.28e21T^{2} \)
89 \( 1 + 6.96e10T + 2.77e21T^{2} \)
97 \( 1 - 2.33e10T + 7.15e21T^{2} \)
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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

−16.73750429946503667993551495141, −15.87906609781069575280976312103, −14.01721976863694618361139233977, −12.20243374244652704098083399569, −10.00772480036113087759238202734, −9.592079496797552020896706665393, −7.995520899264134392699618666307, −6.49390758188442883354091661666, −2.92046330832211970092033260221, −0.981473718487176865459024094512, 0.981473718487176865459024094512, 2.92046330832211970092033260221, 6.49390758188442883354091661666, 7.995520899264134392699618666307, 9.592079496797552020896706665393, 10.00772480036113087759238202734, 12.20243374244652704098083399569, 14.01721976863694618361139233977, 15.87906609781069575280976312103, 16.73750429946503667993551495141

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