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  − 1.30·2-s − 0.302·4-s + 5-s + 7-s + 3·8-s − 1.30·10-s + 3·11-s − 2.30·13-s − 1.30·14-s − 3.30·16-s + 1.30·17-s + 0.302·19-s − 0.302·20-s − 3.90·22-s + 4.30·23-s + 25-s + 3·26-s − 0.302·28-s + 1.69·29-s − 9.60·31-s − 1.69·32-s − 1.69·34-s + 35-s − 3.60·37-s − 0.394·38-s + 3·40-s + 3.90·41-s + ⋯
L(s)  = 1  − 0.921·2-s − 0.151·4-s + 0.447·5-s + 0.377·7-s + 1.06·8-s − 0.411·10-s + 0.904·11-s − 0.638·13-s − 0.348·14-s − 0.825·16-s + 0.315·17-s + 0.0694·19-s − 0.0677·20-s − 0.833·22-s + 0.897·23-s + 0.200·25-s + 0.588·26-s − 0.0572·28-s + 0.315·29-s − 1.72·31-s − 0.300·32-s − 0.291·34-s + 0.169·35-s − 0.592·37-s − 0.0639·38-s + 0.474·40-s + 0.610·41-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  =  $(2,\ 945,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.03386$
$L(\frac12)$  $\approx$  $1.03386$
$L(\frac{3}{2})$   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 \( 1 + 1.30T + 2T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 2.30T + 13T^{2} \)
17 \( 1 - 1.30T + 17T^{2} \)
19 \( 1 - 0.302T + 19T^{2} \)
23 \( 1 - 4.30T + 23T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + 9.60T + 31T^{2} \)
37 \( 1 + 3.60T + 37T^{2} \)
41 \( 1 - 3.90T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 - 0.394T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 8.21T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 + 4.51T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 4.21T + 73T^{2} \)
79 \( 1 + 6.09T + 79T^{2} \)
83 \( 1 - 2.21T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 8.51T + 97T^{2} \)
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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

−9.861698593090997841796095439340, −9.157573622441697866832827370646, −8.671805266307598352259911950727, −7.54840750603366428197244184178, −6.99399019176353753734224880731, −5.69420788193719833106852752689, −4.80938751146346007031929738094, −3.76206591111235032074363160548, −2.16773271275343725080656488029, −0.978012379483523488823020879447, 0.978012379483523488823020879447, 2.16773271275343725080656488029, 3.76206591111235032074363160548, 4.80938751146346007031929738094, 5.69420788193719833106852752689, 6.99399019176353753734224880731, 7.54840750603366428197244184178, 8.671805266307598352259911950727, 9.157573622441697866832827370646, 9.861698593090997841796095439340

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