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  + 72.6·2-s + 243·3-s + 3.22e3·4-s + 3.12e3·5-s + 1.76e4·6-s + 8.97e3·7-s + 8.58e4·8-s + 5.90e4·9-s + 2.27e5·10-s + 4.21e5·11-s + 7.84e5·12-s − 1.24e6·13-s + 6.51e5·14-s + 7.59e5·15-s − 3.78e5·16-s − 5.93e6·17-s + 4.28e6·18-s + 1.88e7·19-s + 1.00e7·20-s + 2.18e6·21-s + 3.05e7·22-s − 2.72e7·23-s + 2.08e7·24-s + 9.76e6·25-s − 9.03e7·26-s + 1.43e7·27-s + 2.89e7·28-s + ⋯
L(s)  = 1  + 1.60·2-s + 0.577·3-s + 1.57·4-s + 0.447·5-s + 0.926·6-s + 0.201·7-s + 0.926·8-s + 0.333·9-s + 0.717·10-s + 0.788·11-s + 0.910·12-s − 0.928·13-s + 0.323·14-s + 0.258·15-s − 0.0901·16-s − 1.01·17-s + 0.535·18-s + 1.74·19-s + 0.705·20-s + 0.116·21-s + 1.26·22-s − 0.882·23-s + 0.534·24-s + 0.199·25-s − 1.49·26-s + 0.192·27-s + 0.318·28-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  =  $(2,\ 15,\ (\ :11/2),\ 1)$
$L(6)$  $\approx$  $5.16168$
$L(\frac12)$  $\approx$  $5.16168$
$L(\frac{13}{2})$   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 \( 1 - 72.6T + 2.04e3T^{2} \)
7 \( 1 - 8.97e3T + 1.97e9T^{2} \)
11 \( 1 - 4.21e5T + 2.85e11T^{2} \)
13 \( 1 + 1.24e6T + 1.79e12T^{2} \)
17 \( 1 + 5.93e6T + 3.42e13T^{2} \)
19 \( 1 - 1.88e7T + 1.16e14T^{2} \)
23 \( 1 + 2.72e7T + 9.52e14T^{2} \)
29 \( 1 + 1.15e8T + 1.22e16T^{2} \)
31 \( 1 + 2.73e8T + 2.54e16T^{2} \)
37 \( 1 + 2.86e8T + 1.77e17T^{2} \)
41 \( 1 - 4.51e8T + 5.50e17T^{2} \)
43 \( 1 - 1.30e9T + 9.29e17T^{2} \)
47 \( 1 + 9.11e8T + 2.47e18T^{2} \)
53 \( 1 - 3.44e9T + 9.26e18T^{2} \)
59 \( 1 - 1.06e10T + 3.01e19T^{2} \)
61 \( 1 - 6.85e9T + 4.35e19T^{2} \)
67 \( 1 - 5.26e9T + 1.22e20T^{2} \)
71 \( 1 - 1.72e10T + 2.31e20T^{2} \)
73 \( 1 - 5.86e9T + 3.13e20T^{2} \)
79 \( 1 + 2.16e10T + 7.47e20T^{2} \)
83 \( 1 + 6.48e9T + 1.28e21T^{2} \)
89 \( 1 - 7.32e10T + 2.77e21T^{2} \)
97 \( 1 + 1.45e11T + 7.15e21T^{2} \)
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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

−16.10754142804007006521952847602, −14.71929773106409637050618574288, −14.01271793697483430622556587521, −12.81831625076203930733094673022, −11.49595669013411873873776368447, −9.399568815553948585832012583558, −7.10949755406581413442626098762, −5.39161432704293351256383159161, −3.81134194280549668901374874555, −2.16047363836934490419594198924, 2.16047363836934490419594198924, 3.81134194280549668901374874555, 5.39161432704293351256383159161, 7.10949755406581413442626098762, 9.399568815553948585832012583558, 11.49595669013411873873776368447, 12.81831625076203930733094673022, 14.01271793697483430622556587521, 14.71929773106409637050618574288, 16.10754142804007006521952847602

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