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  + 1.77·2-s + 1.14·4-s − 0.589·5-s + 7-s − 1.52·8-s − 1.04·10-s + 1.54·13-s + 1.77·14-s − 4.98·16-s + 7.14·17-s + 6.19·19-s − 0.672·20-s − 3.76·23-s − 4.65·25-s + 2.73·26-s + 1.14·28-s + 0.607·29-s − 6.87·31-s − 5.78·32-s + 12.6·34-s − 0.589·35-s − 7.70·37-s + 10.9·38-s + 0.898·40-s + 7.48·41-s + 10.9·43-s − 6.66·46-s + ⋯
L(s)  = 1  + 1.25·2-s + 0.570·4-s − 0.263·5-s + 0.377·7-s − 0.538·8-s − 0.330·10-s + 0.428·13-s + 0.473·14-s − 1.24·16-s + 1.73·17-s + 1.42·19-s − 0.150·20-s − 0.784·23-s − 0.930·25-s + 0.536·26-s + 0.215·28-s + 0.112·29-s − 1.23·31-s − 1.02·32-s + 2.17·34-s − 0.0997·35-s − 1.26·37-s + 1.77·38-s + 0.142·40-s + 1.16·41-s + 1.67·43-s − 0.983·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.852005176$
$L(\frac12)$  $\approx$  $3.852005176$
$L(\frac{3}{2})$   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 \( 1 - 1.77T + 2T^{2} \)
5 \( 1 + 0.589T + 5T^{2} \)
13 \( 1 - 1.54T + 13T^{2} \)
17 \( 1 - 7.14T + 17T^{2} \)
19 \( 1 - 6.19T + 19T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 - 0.607T + 29T^{2} \)
31 \( 1 + 6.87T + 31T^{2} \)
37 \( 1 + 7.70T + 37T^{2} \)
41 \( 1 - 7.48T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + 7.04T + 53T^{2} \)
59 \( 1 - 3.35T + 59T^{2} \)
61 \( 1 - 3.37T + 61T^{2} \)
67 \( 1 + 2.19T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 - 2.58T + 73T^{2} \)
79 \( 1 - 4.66T + 79T^{2} \)
83 \( 1 + 3.50T + 83T^{2} \)
89 \( 1 + 8.15T + 89T^{2} \)
97 \( 1 + 4.79T + 97T^{2} \)
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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

−7.63109544423908778019810385569, −7.26661683609046260915398475538, −6.10757512079227294252983481102, −5.58494536198920968599180602832, −5.22525432472344898664076075978, −4.15956440382082611091685504089, −3.70066824518540539816468402810, −3.05474524299489968316174172664, −2.00519686638820252879694456824, −0.824686871640981484672174920591, 0.824686871640981484672174920591, 2.00519686638820252879694456824, 3.05474524299489968316174172664, 3.70066824518540539816468402810, 4.15956440382082611091685504089, 5.22525432472344898664076075978, 5.58494536198920968599180602832, 6.10757512079227294252983481102, 7.26661683609046260915398475538, 7.63109544423908778019810385569

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