Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s − 3·4-s + 5·5-s − 3·6-s − 7·8-s + 9·9-s + 5·10-s + 9·12-s − 15·15-s + 5·16-s − 14·17-s + 9·18-s − 22·19-s − 15·20-s + 34·23-s + 21·24-s + 25·25-s − 27·27-s − 15·30-s + 2·31-s + 33·32-s − 14·34-s − 27·36-s − 22·38-s − 35·40-s + 45·45-s + ⋯
L(s)  = 1  + 1/2·2-s − 3-s − 3/4·4-s + 5-s − 1/2·6-s − 7/8·8-s + 9-s + 1/2·10-s + 3/4·12-s − 15-s + 5/16·16-s − 0.823·17-s + 1/2·18-s − 1.15·19-s − 3/4·20-s + 1.47·23-s + 7/8·24-s + 25-s − 27-s − 1/2·30-s + 2/31·31-s + 1.03·32-s − 0.411·34-s − 3/4·36-s − 0.578·38-s − 7/8·40-s + 45-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  $\chi_{15} (14, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 15,\ (\ :1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.747644$
$L(\frac12)$  $\approx$  $0.747644$
$L(2)$   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 \( 1 + p T \)
5 \( 1 - p T \)
good2 \( 1 - T + p^{2} T^{2} \)
7 \( ( 1 - p T )( 1 + p T ) \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( 1 + 14 T + p^{2} T^{2} \)
19 \( 1 + 22 T + p^{2} T^{2} \)
23 \( 1 - 34 T + p^{2} T^{2} \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 2 T + p^{2} T^{2} \)
37 \( ( 1 - p T )( 1 + p T ) \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( ( 1 - p T )( 1 + p T ) \)
47 \( 1 + 14 T + p^{2} T^{2} \)
53 \( 1 + 86 T + p^{2} T^{2} \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 + 118 T + p^{2} T^{2} \)
67 \( ( 1 - p T )( 1 + p T ) \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( 1 - 98 T + p^{2} T^{2} \)
83 \( 1 - 154 T + p^{2} T^{2} \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( ( 1 - p T )( 1 + p T ) \)
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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

−18.84216747882218125052134749858, −17.73632491044169124692694506319, −16.94133112513282712743435662192, −15.07325242580970539835333354209, −13.50572692666400849793767943581, −12.59137691919670652128806918476, −10.74164239244868709362486626799, −9.195310302182541695476289542121, −6.33181250203772271694008358554, −4.84192581422996258804553371125, 4.84192581422996258804553371125, 6.33181250203772271694008358554, 9.195310302182541695476289542121, 10.74164239244868709362486626799, 12.59137691919670652128806918476, 13.50572692666400849793767943581, 15.07325242580970539835333354209, 16.94133112513282712743435662192, 17.73632491044169124692694506319, 18.84216747882218125052134749858

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