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-s − 4-s + 2·5-s + 7-s + 3·8-s − 2·10-s + 2·13-s − 14-s − 16-s − 6·17-s − 4·19-s − 2·20-s − 25-s − 2·26-s − 28-s − 2·29-s − 5·32-s + 6·34-s + 2·35-s + 6·37-s + 4·38-s + 6·40-s + 2·41-s + 4·43-s + 49-s + 50-s − 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 0.632·10-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s − 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s − 0.883·32-s + 1.02·34-s + 0.338·35-s + 0.986·37-s + 0.648·38-s + 0.948·40-s + 0.312·41-s + 0.609·43-s + 1/7·49-s + 0.141·50-s − 0.277·52-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.256465797$
$L(\frac12)$  $\approx$  $1.256465797$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−17.14055573366832, −16.83575571401204, −15.93027713845859, −15.40965513850226, −14.61498799125813, −14.03507831529025, −13.53481514762512, −13.02391570779124, −12.53968112568611, −11.34220857640261, −11.01049711886279, −10.36427958838071, −9.730563286935959, −9.056096586309823, −8.783200717465471, −7.976525779912434, −7.393988504131659, −6.345180936532752, −6.015810616969177, −4.924035533657760, −4.467926406572259, −3.620887797848250, −2.280591623076023, −1.776112969349438, −0.6333466336966226, 0.6333466336966226, 1.776112969349438, 2.280591623076023, 3.620887797848250, 4.467926406572259, 4.924035533657760, 6.015810616969177, 6.345180936532752, 7.393988504131659, 7.976525779912434, 8.783200717465471, 9.056096586309823, 9.730563286935959, 10.36427958838071, 11.01049711886279, 11.34220857640261, 12.53968112568611, 13.02391570779124, 13.53481514762512, 14.03507831529025, 14.61498799125813, 15.40965513850226, 15.93027713845859, 16.83575571401204, 17.14055573366832

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