Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.46·2-s + 0.151·4-s − 0.466·5-s − 7-s − 2.71·8-s − 0.684·10-s − 1.58·13-s − 1.46·14-s − 4.27·16-s − 5.22·17-s + 4.22·19-s − 0.0706·20-s + 1.80·23-s − 4.78·25-s − 2.32·26-s − 0.151·28-s − 2.71·29-s + 1.29·31-s − 0.854·32-s − 7.66·34-s + 0.466·35-s − 1.94·37-s + 6.19·38-s + 1.26·40-s + 1.04·41-s + 8.70·43-s + 2.64·46-s + ⋯
L(s)  = 1  + 1.03·2-s + 0.0756·4-s − 0.208·5-s − 0.377·7-s − 0.958·8-s − 0.216·10-s − 0.438·13-s − 0.392·14-s − 1.06·16-s − 1.26·17-s + 0.968·19-s − 0.0157·20-s + 0.376·23-s − 0.956·25-s − 0.455·26-s − 0.0285·28-s − 0.504·29-s + 0.232·31-s − 0.150·32-s − 1.31·34-s + 0.0788·35-s − 0.319·37-s + 1.00·38-s + 0.200·40-s + 0.162·41-s + 1.32·43-s + 0.390·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7623} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.963328034$
$L(\frac12)$  $\approx$  $1.963328034$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - 1.46T + 2T^{2} \)
5 \( 1 + 0.466T + 5T^{2} \)
13 \( 1 + 1.58T + 13T^{2} \)
17 \( 1 + 5.22T + 17T^{2} \)
19 \( 1 - 4.22T + 19T^{2} \)
23 \( 1 - 1.80T + 23T^{2} \)
29 \( 1 + 2.71T + 29T^{2} \)
31 \( 1 - 1.29T + 31T^{2} \)
37 \( 1 + 1.94T + 37T^{2} \)
41 \( 1 - 1.04T + 41T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 - 6.39T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 - 8.60T + 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 + 4.67T + 67T^{2} \)
71 \( 1 + 9.74T + 71T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 - 3.58T + 79T^{2} \)
83 \( 1 - 17.2T + 83T^{2} \)
89 \( 1 - 8.91T + 89T^{2} \)
97 \( 1 + 2.70T + 97T^{2} \)
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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

−7.56067269434262933519432683879, −7.20491024354777256895295165105, −6.19991915356414359159739705089, −5.77245154566579894196991570263, −4.94910520429000959012218499147, −4.31999238814034425015889028604, −3.69172126476643500723740512163, −2.89272923623505996682177100034, −2.12892942546241648956504482384, −0.57279222918105059945122563427, 0.57279222918105059945122563427, 2.12892942546241648956504482384, 2.89272923623505996682177100034, 3.69172126476643500723740512163, 4.31999238814034425015889028604, 4.94910520429000959012218499147, 5.77245154566579894196991570263, 6.19991915356414359159739705089, 7.20491024354777256895295165105, 7.56067269434262933519432683879

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