Properties

Degree $2$
Conductor $67760$
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 − 7-s − 3·9-s + 6·13-s − 2·17-s + 25-s − 6·29-s − 8·31-s + 35-s − 10·37-s − 2·41-s + 4·43-s + 3·45-s − 8·47-s + 49-s − 2·53-s + 8·59-s + 14·61-s + 3·63-s − 6·65-s + 12·67-s + 16·71-s − 2·73-s − 8·79-s + 9·81-s + 8·83-s + 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 9-s + 1.66·13-s − 0.485·17-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.169·35-s − 1.64·37-s − 0.312·41-s + 0.609·43-s + 0.447·45-s − 1.16·47-s + 1/7·49-s − 0.274·53-s + 1.04·59-s + 1.79·61-s + 0.377·63-s − 0.744·65-s + 1.46·67-s + 1.89·71-s − 0.234·73-s − 0.900·79-s + 81-s + 0.878·83-s + 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(67760\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{67760} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 67760,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.39179770374418, −13.98848414167809, −13.33672383385777, −12.95956990348462, −12.55898013554595, −11.65407698142846, −11.49309203886006, −10.89617834678969, −10.64271261187195, −9.802951912850788, −9.180794135076855, −8.811886045896159, −8.305452490685750, −7.910036997970959, −7.031148602582290, −6.686060707056571, −6.076392013508345, −5.414490075977663, −5.160042131059455, −4.063127064142139, −3.561700251758383, −3.389344447717153, −2.343148855701056, −1.774226463801084, −0.7787978553525388, 0, 0.7787978553525388, 1.774226463801084, 2.343148855701056, 3.389344447717153, 3.561700251758383, 4.063127064142139, 5.160042131059455, 5.414490075977663, 6.076392013508345, 6.686060707056571, 7.031148602582290, 7.910036997970959, 8.305452490685750, 8.811886045896159, 9.180794135076855, 9.802951912850788, 10.64271261187195, 10.89617834678969, 11.49309203886006, 11.65407698142846, 12.55898013554595, 12.95956990348462, 13.33672383385777, 13.98848414167809, 14.39179770374418

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