Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \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  + 2·5-s + 4·7-s + 2·13-s − 2·17-s + 8·23-s − 25-s − 6·29-s + 8·31-s + 8·35-s + 6·37-s − 2·41-s + 8·47-s + 9·49-s − 6·53-s − 4·59-s − 6·61-s + 4·65-s + 4·67-s + 14·73-s − 4·79-s − 12·83-s − 4·85-s + 6·89-s + 8·91-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s + 0.554·13-s − 0.485·17-s + 1.66·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 1.35·35-s + 0.986·37-s − 0.312·41-s + 1.16·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s − 0.768·61-s + 0.496·65-s + 0.488·67-s + 1.63·73-s − 0.450·79-s − 1.31·83-s − 0.433·85-s + 0.635·89-s + 0.838·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(17424\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{17424} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 17424,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.822234730$
$L(\frac12)$  $\approx$  $3.822234730$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−15.62830070356156, −15.26409721674001, −14.78409630668018, −14.00951033870826, −13.83820762124744, −13.14383097729821, −12.67692582912470, −11.80866906021705, −11.25329400604665, −10.96239468866132, −10.32348465462530, −9.552101696182188, −9.067769145208257, −8.477095439585001, −7.880921482514082, −7.288578518721846, −6.496320961370779, −5.909164388335791, −5.247599180478130, −4.713583447197854, −4.086245959478811, −3.056907216238127, −2.267752673328799, −1.602120980589437, −0.8904474913341941, 0.8904474913341941, 1.602120980589437, 2.267752673328799, 3.056907216238127, 4.086245959478811, 4.713583447197854, 5.247599180478130, 5.909164388335791, 6.496320961370779, 7.288578518721846, 7.880921482514082, 8.477095439585001, 9.067769145208257, 9.552101696182188, 10.32348465462530, 10.96239468866132, 11.25329400604665, 11.80866906021705, 12.67692582912470, 13.14383097729821, 13.83820762124744, 14.00951033870826, 14.78409630668018, 15.26409721674001, 15.62830070356156

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