Properties

Degree 2
Conductor $ 3 \cdot 5 $
Sign $1$
Motivic weight 7
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.75·2-s + 27·3-s − 51.3·4-s + 125·5-s − 236.·6-s + 1.33e3·7-s + 1.57e3·8-s + 729·9-s − 1.09e3·10-s + 7.41e3·11-s − 1.38e3·12-s − 1.45e4·13-s − 1.17e4·14-s + 3.37e3·15-s − 7.18e3·16-s + 1.44e4·17-s − 6.38e3·18-s + 409.·19-s − 6.41e3·20-s + 3.61e4·21-s − 6.49e4·22-s − 1.79e4·23-s + 4.23e4·24-s + 1.56e4·25-s + 1.27e5·26-s + 1.96e4·27-s − 6.86e4·28-s + ⋯
L(s)  = 1  − 0.774·2-s + 0.577·3-s − 0.400·4-s + 0.447·5-s − 0.446·6-s + 1.47·7-s + 1.08·8-s + 0.333·9-s − 0.346·10-s + 1.67·11-s − 0.231·12-s − 1.84·13-s − 1.14·14-s + 0.258·15-s − 0.438·16-s + 0.711·17-s − 0.258·18-s + 0.0137·19-s − 0.179·20-s + 0.851·21-s − 1.29·22-s − 0.307·23-s + 0.626·24-s + 0.199·25-s + 1.42·26-s + 0.192·27-s − 0.591·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{15} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 15,\ (\ :7/2),\ 1)$
$L(4)$  $\approx$  $1.34799$
$L(\frac12)$  $\approx$  $1.34799$
$L(\frac{9}{2})$   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 \( 1 - 27T \)
5 \( 1 - 125T \)
good2 \( 1 + 8.75T + 128T^{2} \)
7 \( 1 - 1.33e3T + 8.23e5T^{2} \)
11 \( 1 - 7.41e3T + 1.94e7T^{2} \)
13 \( 1 + 1.45e4T + 6.27e7T^{2} \)
17 \( 1 - 1.44e4T + 4.10e8T^{2} \)
19 \( 1 - 409.T + 8.93e8T^{2} \)
23 \( 1 + 1.79e4T + 3.40e9T^{2} \)
29 \( 1 + 1.07e4T + 1.72e10T^{2} \)
31 \( 1 - 1.78e5T + 2.75e10T^{2} \)
37 \( 1 + 4.27e5T + 9.49e10T^{2} \)
41 \( 1 - 5.33e4T + 1.94e11T^{2} \)
43 \( 1 - 8.93e4T + 2.71e11T^{2} \)
47 \( 1 + 1.61e5T + 5.06e11T^{2} \)
53 \( 1 + 1.21e5T + 1.17e12T^{2} \)
59 \( 1 + 1.69e6T + 2.48e12T^{2} \)
61 \( 1 + 1.23e6T + 3.14e12T^{2} \)
67 \( 1 + 9.44e5T + 6.06e12T^{2} \)
71 \( 1 + 9.36e5T + 9.09e12T^{2} \)
73 \( 1 + 5.49e6T + 1.10e13T^{2} \)
79 \( 1 + 3.29e6T + 1.92e13T^{2} \)
83 \( 1 + 4.16e6T + 2.71e13T^{2} \)
89 \( 1 - 8.50e6T + 4.42e13T^{2} \)
97 \( 1 - 6.61e6T + 8.07e13T^{2} \)
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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

−17.55842006008268944748005967664, −17.04877873582142923487452575919, −14.62274834405699445967999911824, −14.07335447537991267461008271188, −11.99031239457933918155157372866, −10.06152442372522655314365043787, −8.895535089709248355697539610103, −7.54029388453871757024335302251, −4.66131089137715335268124592628, −1.55323263000999941611920540855, 1.55323263000999941611920540855, 4.66131089137715335268124592628, 7.54029388453871757024335302251, 8.895535089709248355697539610103, 10.06152442372522655314365043787, 11.99031239457933918155157372866, 14.07335447537991267461008271188, 14.62274834405699445967999911824, 17.04877873582142923487452575919, 17.55842006008268944748005967664

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