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  + 2·2-s + 2·4-s − 5-s − 7-s − 2·10-s + 4·13-s − 2·14-s − 4·16-s − 17-s − 2·20-s + 4·23-s − 4·25-s + 8·26-s − 2·28-s − 2·31-s − 8·32-s − 2·34-s + 35-s + 6·37-s + 2·41-s + 3·43-s + 8·46-s + 7·47-s + 49-s − 8·50-s + 8·52-s + 12·53-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.447·5-s − 0.377·7-s − 0.632·10-s + 1.10·13-s − 0.534·14-s − 16-s − 0.242·17-s − 0.447·20-s + 0.834·23-s − 4/5·25-s + 1.56·26-s − 0.377·28-s − 0.359·31-s − 1.41·32-s − 0.342·34-s + 0.169·35-s + 0.986·37-s + 0.312·41-s + 0.457·43-s + 1.17·46-s + 1.02·47-s + 1/7·49-s − 1.13·50-s + 1.10·52-s + 1.64·53-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  :  $\chi_{7623} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.680537577$
$L(\frac12)$  $\approx$  $3.680537577$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−16.82146344245696, −16.23078593988663, −15.57885749167578, −15.39144097279012, −14.60676082275961, −14.08838393630710, −13.39435009871519, −13.06915372058273, −12.51773765344947, −11.74562798394210, −11.35535911201250, −10.75308077427911, −9.907195909860546, −9.025530622955453, −8.635201561111230, −7.615718676839990, −7.017853977652872, −6.193178258578768, −5.796396613005271, −5.004883721472250, −4.130417876790426, −3.784195323697624, −2.990058312386353, −2.162929059422237, −0.7630901429310847, 0.7630901429310847, 2.162929059422237, 2.990058312386353, 3.784195323697624, 4.130417876790426, 5.004883721472250, 5.796396613005271, 6.193178258578768, 7.017853977652872, 7.615718676839990, 8.635201561111230, 9.025530622955453, 9.907195909860546, 10.75308077427911, 11.35535911201250, 11.74562798394210, 12.51773765344947, 13.06915372058273, 13.39435009871519, 14.08838393630710, 14.60676082275961, 15.39144097279012, 15.57885749167578, 16.23078593988663, 16.82146344245696

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