Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 11 \cdot 13^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 2·7-s + 9-s − 11-s − 2·15-s + 4·17-s + 6·19-s + 2·21-s − 25-s + 27-s − 8·29-s + 8·31-s − 33-s − 4·35-s − 10·37-s − 8·41-s − 2·43-s − 2·45-s + 8·47-s − 3·49-s + 4·51-s − 2·53-s + 2·55-s + 6·57-s − 12·59-s + 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.516·15-s + 0.970·17-s + 1.37·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.174·33-s − 0.676·35-s − 1.64·37-s − 1.24·41-s − 0.304·43-s − 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.269·55-s + 0.794·57-s − 1.56·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(22308\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{22308} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 22308,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
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

−15.62271680277025, −15.34956178131870, −14.68628346992860, −14.23765179220349, −13.63791030782462, −13.29368245332553, −12.28812570638841, −12.02627933194238, −11.52572478669403, −10.94696180975043, −10.12899821196670, −9.863305636923337, −8.953040184581630, −8.506844971464490, −7.855515273734475, −7.504732576521906, −7.090639442664263, −6.032640799435036, −5.344520226898016, −4.815126687581783, −4.063838589543664, −3.382334714397497, −2.959546418779256, −1.841406346845498, −1.213637230308183, 0, 1.213637230308183, 1.841406346845498, 2.959546418779256, 3.382334714397497, 4.063838589543664, 4.815126687581783, 5.344520226898016, 6.032640799435036, 7.090639442664263, 7.504732576521906, 7.855515273734475, 8.506844971464490, 8.953040184581630, 9.863305636923337, 10.12899821196670, 10.94696180975043, 11.52572478669403, 12.02627933194238, 12.28812570638841, 13.29368245332553, 13.63791030782462, 14.23765179220349, 14.68628346992860, 15.34956178131870, 15.62271680277025

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