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Properties

Label 2-720-12.11-c5-0-8
Degree $2$
Conductor $720$
Sign $0.0917 - 0.995i$
Analytic cond. $115.476$
Root an. cond. $10.7459$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25i·5-s − 167. i·7-s + 184.·11-s − 234.·13-s + 1.22e3i·17-s − 965. i·19-s − 2.55e3·23-s − 625·25-s + 4.13e3i·29-s − 530. i·31-s + 4.18e3·35-s − 9.45e3·37-s − 1.37e3i·41-s − 9.52e3i·43-s + 3.65e3·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.29i·7-s + 0.460·11-s − 0.385·13-s + 1.02i·17-s − 0.613i·19-s − 1.00·23-s − 0.200·25-s + 0.913i·29-s − 0.0990i·31-s + 0.578·35-s − 1.13·37-s − 0.127i·41-s − 0.785i·43-s + 0.241·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.0917 - 0.995i$
Analytic conductor: \(115.476\)
Root analytic conductor: \(10.7459\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :5/2),\ 0.0917 - 0.995i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.266017551\)
\(L(\frac12)\) \(\approx\) \(1.266017551\)
\(L(\frac{7}{2})\) 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 \( 1 - 25iT \)
good7 \( 1 + 167. iT - 1.68e4T^{2} \)
11 \( 1 - 184.T + 1.61e5T^{2} \)
13 \( 1 + 234.T + 3.71e5T^{2} \)
17 \( 1 - 1.22e3iT - 1.41e6T^{2} \)
19 \( 1 + 965. iT - 2.47e6T^{2} \)
23 \( 1 + 2.55e3T + 6.43e6T^{2} \)
29 \( 1 - 4.13e3iT - 2.05e7T^{2} \)
31 \( 1 + 530. iT - 2.86e7T^{2} \)
37 \( 1 + 9.45e3T + 6.93e7T^{2} \)
41 \( 1 + 1.37e3iT - 1.15e8T^{2} \)
43 \( 1 + 9.52e3iT - 1.47e8T^{2} \)
47 \( 1 - 3.65e3T + 2.29e8T^{2} \)
53 \( 1 - 1.37e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.86e4T + 7.14e8T^{2} \)
61 \( 1 - 4.03e4T + 8.44e8T^{2} \)
67 \( 1 - 4.03e3iT - 1.35e9T^{2} \)
71 \( 1 - 2.58e4T + 1.80e9T^{2} \)
73 \( 1 - 2.48e4T + 2.07e9T^{2} \)
79 \( 1 - 7.71e4iT - 3.07e9T^{2} \)
83 \( 1 + 3.82e4T + 3.93e9T^{2} \)
89 \( 1 + 6.42e4iT - 5.58e9T^{2} \)
97 \( 1 - 4.81e4T + 8.58e9T^{2} \)
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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

−10.11327641928905990025635183359, −9.003844167566999188209652649401, −8.031220419517944231423666033537, −7.12376560642611934191190383792, −6.57631957932848218178780131508, −5.37672022815468866485325414720, −4.16001976742402005326340584957, −3.55366675165208250892205019595, −2.13889828813934820204872785735, −0.949664398485628919816153307715, 0.29579171007640814660054596199, 1.74556365993955698291370775963, 2.67295636995520341052049692181, 3.93693389991475598354128907545, 5.06632687698063015714958999238, 5.76716580706783350328197140781, 6.72955720372225820414122551012, 7.889678529520331626296861329655, 8.622564990498740543659218112309, 9.443059848895988899174595067063

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