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  + 1.61·2-s + 0.618·4-s + 2.23·5-s − 7-s − 2.23·8-s + 3.61·10-s − 0.236·13-s − 1.61·14-s − 4.85·16-s + 2·17-s + 1.47·19-s + 1.38·20-s + 2·23-s − 0.381·26-s − 0.618·28-s + 5·29-s + 10.4·31-s − 3.38·32-s + 3.23·34-s − 2.23·35-s − 1.47·37-s + 2.38·38-s − 5.00·40-s + 2.47·41-s + 8.47·43-s + 3.23·46-s − 9.47·47-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.309·4-s + 0.999·5-s − 0.377·7-s − 0.790·8-s + 1.14·10-s − 0.0654·13-s − 0.432·14-s − 1.21·16-s + 0.485·17-s + 0.337·19-s + 0.309·20-s + 0.417·23-s − 0.0749·26-s − 0.116·28-s + 0.928·29-s + 1.88·31-s − 0.597·32-s + 0.554·34-s − 0.377·35-s − 0.242·37-s + 0.386·38-s − 0.790·40-s + 0.386·41-s + 1.29·43-s + 0.477·46-s − 1.38·47-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.966334855$
$L(\frac12)$  $\approx$  $3.966334855$
$L(\frac{3}{2})$   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 \( 1 - 1.61T + 2T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
13 \( 1 + 0.236T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 1.47T + 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + 1.47T + 37T^{2} \)
41 \( 1 - 2.47T + 41T^{2} \)
43 \( 1 - 8.47T + 43T^{2} \)
47 \( 1 + 9.47T + 47T^{2} \)
53 \( 1 + 6.47T + 53T^{2} \)
59 \( 1 + 7.94T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 9.18T + 67T^{2} \)
71 \( 1 - 2.47T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 0.472T + 79T^{2} \)
83 \( 1 - 7.52T + 83T^{2} \)
89 \( 1 - 0.472T + 89T^{2} \)
97 \( 1 - 9.41T + 97T^{2} \)
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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

−7.81831357650664007853897358258, −6.82764814858202645336872175834, −6.16340696168307906343622535329, −5.88323085076054928860637490054, −4.89890866966525158300282345702, −4.58978580035627707297643587465, −3.44453051566755071249069025030, −2.93850217898377106423188940653, −2.08770197072324305690088884815, −0.840012123146977486997558457768, 0.840012123146977486997558457768, 2.08770197072324305690088884815, 2.93850217898377106423188940653, 3.44453051566755071249069025030, 4.58978580035627707297643587465, 4.89890866966525158300282345702, 5.88323085076054928860637490054, 6.16340696168307906343622535329, 6.82764814858202645336872175834, 7.81831357650664007853897358258

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