Properties

Degree $2$
Conductor $360789$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 2·5-s − 6-s − 3·8-s + 9-s − 2·10-s + 11-s + 12-s + 13-s + 2·15-s − 16-s + 6·17-s + 18-s + 4·19-s + 2·20-s + 22-s − 8·23-s + 3·24-s − 25-s + 26-s − 27-s + 2·30-s + 5·32-s − 33-s + 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.516·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.213·22-s − 1.66·23-s + 0.612·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.365·30-s + 0.883·32-s − 0.174·33-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360789 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360789\)    =    \(3 \cdot 11 \cdot 13 \cdot 29^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{360789} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 360789,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
29 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.61529701280458, −12.18000874326554, −11.95230988834057, −11.64115009737789, −11.18778820417822, −10.44914347259765, −9.962717684170374, −9.779194557837950, −9.129258679360143, −8.575079179615233, −8.085562280586856, −7.658215356353374, −7.374805849758219, −6.427883792772483, −6.187191563566249, −5.778350903899441, −5.067684641208160, −4.808336114326416, −4.349135521801371, −3.580033871411205, −3.354399257001126, −3.134376012069771, −1.781072258605091, −1.515207136911775, −0.5236684833473974, 0, 0.5236684833473974, 1.515207136911775, 1.781072258605091, 3.134376012069771, 3.354399257001126, 3.580033871411205, 4.349135521801371, 4.808336114326416, 5.067684641208160, 5.778350903899441, 6.187191563566249, 6.427883792772483, 7.374805849758219, 7.658215356353374, 8.085562280586856, 8.575079179615233, 9.129258679360143, 9.779194557837950, 9.962717684170374, 10.44914347259765, 11.18778820417822, 11.64115009737789, 11.95230988834057, 12.18000874326554, 12.61529701280458

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