Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s − 1.82·4-s − 5-s − 7-s − 1.58·8-s − 0.414·10-s + 2.41·11-s − 0.414·13-s − 0.414·14-s + 3·16-s + 0.828·17-s + 4.41·19-s + 1.82·20-s + 0.999·22-s − 4.82·23-s + 25-s − 0.171·26-s + 1.82·28-s + 4·29-s + 6·31-s + 4.41·32-s + 0.343·34-s + 35-s + 8.48·37-s + 1.82·38-s + 1.58·40-s + 2.17·41-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.914·4-s − 0.447·5-s − 0.377·7-s − 0.560·8-s − 0.130·10-s + 0.727·11-s − 0.114·13-s − 0.110·14-s + 0.750·16-s + 0.200·17-s + 1.01·19-s + 0.408·20-s + 0.213·22-s − 1.00·23-s + 0.200·25-s − 0.0336·26-s + 0.345·28-s + 0.742·29-s + 1.07·31-s + 0.780·32-s + 0.0588·34-s + 0.169·35-s + 1.39·37-s + 0.296·38-s + 0.250·40-s + 0.339·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{945} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 945,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.27025$
$L(\frac12)$  $\approx$  $1.27025$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
good2 \( 1 - 0.414T + 2T^{2} \)
11 \( 1 - 2.41T + 11T^{2} \)
13 \( 1 + 0.414T + 13T^{2} \)
17 \( 1 - 0.828T + 17T^{2} \)
19 \( 1 - 4.41T + 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 - 2.17T + 41T^{2} \)
43 \( 1 - 4.65T + 43T^{2} \)
47 \( 1 - 9.48T + 47T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 - 2.17T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 9.17T + 79T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 + 2.65T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−9.756276417316992312531199280960, −9.399618859337813692756970664894, −8.349047094470360653597024532452, −7.67075274733825648820178862853, −6.49157910502769604567010324267, −5.67617466705980641713739976336, −4.57129197376866602325310791351, −3.88628529066027400245505538327, −2.86280651394709953847482887950, −0.878116091487073529779634856287, 0.878116091487073529779634856287, 2.86280651394709953847482887950, 3.88628529066027400245505538327, 4.57129197376866602325310791351, 5.67617466705980641713739976336, 6.49157910502769604567010324267, 7.67075274733825648820178862853, 8.349047094470360653597024532452, 9.399618859337813692756970664894, 9.756276417316992312531199280960

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