Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{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  + 5-s − 4·11-s + 2·13-s + 2·17-s − 4·19-s − 4·23-s + 25-s + 2·29-s + 8·31-s + 6·37-s − 6·41-s − 8·43-s + 4·47-s − 6·53-s − 4·55-s − 4·59-s + 2·61-s + 2·65-s + 8·67-s + 6·73-s − 16·83-s + 2·85-s − 6·89-s − 4·95-s + 14·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.986·37-s − 0.937·41-s − 1.21·43-s + 0.583·47-s − 0.824·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s + 0.977·67-s + 0.702·73-s − 1.75·83-s + 0.216·85-s − 0.635·89-s − 0.410·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(17640\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{17640} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 17640,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.96496035694961, −15.58272733772508, −15.15234538353918, −14.29687580022507, −13.96540141409430, −13.25341742073796, −12.95698583838460, −12.28927899855731, −11.65644251418012, −11.05833349758285, −10.29300025660193, −10.14764209619486, −9.452235162029079, −8.548050572972440, −8.209298532389627, −7.682029060615088, −6.739872189722621, −6.261449851666244, −5.655837542192529, −4.957842128239501, −4.362985064743386, −3.479318783498051, −2.728973960701881, −2.090142430852942, −1.140035216622331, 0, 1.140035216622331, 2.090142430852942, 2.728973960701881, 3.479318783498051, 4.362985064743386, 4.957842128239501, 5.655837542192529, 6.261449851666244, 6.739872189722621, 7.682029060615088, 8.209298532389627, 8.548050572972440, 9.452235162029079, 10.14764209619486, 10.29300025660193, 11.05833349758285, 11.65644251418012, 12.28927899855731, 12.95698583838460, 13.25341742073796, 13.96540141409430, 14.29687580022507, 15.15234538353918, 15.58272733772508, 15.96496035694961

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