Properties

Label 2-8640-1.1-c1-0-11
Degree $2$
Conductor $8640$
Sign $1$
Analytic cond. $68.9907$
Root an. cond. $8.30606$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s − 4·11-s + 2·13-s − 5·17-s − 5·19-s + 23-s + 25-s − 2·29-s − 7·31-s − 2·35-s + 6·37-s + 4·43-s + 4·47-s − 3·49-s + 9·53-s − 4·55-s − 14·59-s + 11·61-s + 2·65-s + 14·67-s − 12·73-s + 8·77-s + 3·79-s + 83-s − 5·85-s − 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 1.20·11-s + 0.554·13-s − 1.21·17-s − 1.14·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s − 1.25·31-s − 0.338·35-s + 0.986·37-s + 0.609·43-s + 0.583·47-s − 3/7·49-s + 1.23·53-s − 0.539·55-s − 1.82·59-s + 1.40·61-s + 0.248·65-s + 1.71·67-s − 1.40·73-s + 0.911·77-s + 0.337·79-s + 0.109·83-s − 0.542·85-s − 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8640\)    =    \(2^{6} \cdot 3^{3} \cdot 5\)
Sign: $1$
Analytic conductor: \(68.9907\)
Root analytic conductor: \(8.30606\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.243115331\)
\(L(\frac12)\) \(\approx\) \(1.243115331\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−7.69381804337674001973337841786, −7.06666940321106711649016532162, −6.27971706861172856969557550419, −5.86740150416520331510513417666, −5.02277539706067172285035752952, −4.25072470920032214502565100153, −3.44931407523417704011114555728, −2.51930996021286779208892591113, −1.98338657662642604152391065145, −0.51275164904465627438494634049, 0.51275164904465627438494634049, 1.98338657662642604152391065145, 2.51930996021286779208892591113, 3.44931407523417704011114555728, 4.25072470920032214502565100153, 5.02277539706067172285035752952, 5.86740150416520331510513417666, 6.27971706861172856969557550419, 7.06666940321106711649016532162, 7.69381804337674001973337841786

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