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Properties

Label 2-3240-360.259-c0-0-0
Degree 22
Conductor 32403240
Sign 0.173−0.984i0.173 - 0.984i
Analytic cond. 1.616971.61697
Root an. cond. 1.271601.27160
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−1 + 1.73i)11-s + (−0.5 − 0.866i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 19-s + (0.499 − 0.866i)20-s + (0.999 + 1.73i)22-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s − 0.999·26-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−1 + 1.73i)11-s + (−0.5 − 0.866i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 19-s + (0.499 − 0.866i)20-s + (0.999 + 1.73i)22-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s − 0.999·26-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.173−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s) L(s)=((0.173−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.173−0.984i0.173 - 0.984i
Analytic conductor: 1.616971.61697
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3240(379,⋅)\chi_{3240} (379, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :0), 0.173−0.984i)(2,\ 3240,\ (\ :0),\ 0.173 - 0.984i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.78735385700.7873538570
L(12)L(\frac12) ≈\approx 0.78735385700.7873538570
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1 1
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good7 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
11 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
13 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1+T+T2 1 + T + T^{2}
23 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
29 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+2T+T2 1 + 2T + T^{2}
41 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1−T+T2 1 - T + T^{2}
59 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1−2T+T2 1 - 2T + T^{2}
97 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−9.300241446207699138752368806779, −8.401562012503533660608637083480, −7.38832894674501836911077391259, −6.58574643577698575444650269880, −5.83561951833600399530233948508, −5.13607577752474990027587761519, −4.36600969732535765400587024428, −3.17091040151054590340488845392, −2.45300861189565052257435052963, −2.01363870509096744759729799137, 0.35885198895837900450068662967, 2.15989324418934498254941759176, 3.43739111128036782504130111231, 4.02982758383732957212725432772, 5.06483125179744344767590084824, 5.57494243814410927230452997667, 6.36825306964191468156133545408, 7.05741951983430526880300858984, 7.932854904842347185063128532297, 8.592146186600905207199010545852

Graph of the ZZ-function along the critical line