Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 11-s + 2·13-s + 4·17-s + 6·19-s − 25-s + 8·29-s + 8·31-s + 10·37-s + 8·41-s − 2·43-s − 8·47-s + 2·53-s − 2·55-s + 12·59-s − 10·61-s + 4·65-s + 12·67-s − 8·71-s − 6·73-s − 2·79-s + 16·83-s + 8·85-s − 14·89-s + 12·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.301·11-s + 0.554·13-s + 0.970·17-s + 1.37·19-s − 1/5·25-s + 1.48·29-s + 1.43·31-s + 1.64·37-s + 1.24·41-s − 0.304·43-s − 1.16·47-s + 0.274·53-s − 0.269·55-s + 1.56·59-s − 1.28·61-s + 0.496·65-s + 1.46·67-s − 0.949·71-s − 0.702·73-s − 0.225·79-s + 1.75·83-s + 0.867·85-s − 1.48·89-s + 1.23·95-s + 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19404 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19404\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19404} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 19404,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.477951267$
$L(\frac12)$  $\approx$  $3.477951267$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−15.90668663749456, −15.11138191011875, −14.50253273410853, −13.99395914745715, −13.56107082499534, −13.13248056766643, −12.39775896349132, −11.82931287502949, −11.36555763022687, −10.59274794702250, −9.961455238143301, −9.729222330084138, −9.112476653662201, −8.101752659331624, −8.028552957622371, −7.094302037995701, −6.376812989502171, −5.896040796982790, −5.325417629381837, −4.662276720520456, −3.855221761628750, −2.934841547956909, −2.562237899046148, −1.384532550152447, −0.8741593823422171, 0.8741593823422171, 1.384532550152447, 2.562237899046148, 2.934841547956909, 3.855221761628750, 4.662276720520456, 5.325417629381837, 5.896040796982790, 6.376812989502171, 7.094302037995701, 8.028552957622371, 8.101752659331624, 9.112476653662201, 9.729222330084138, 9.961455238143301, 10.59274794702250, 11.36555763022687, 11.82931287502949, 12.39775896349132, 13.13248056766643, 13.56107082499534, 13.99395914745715, 14.50253273410853, 15.11138191011875, 15.90668663749456

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