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Properties

Label 2-3240-360.259-c0-0-5
Degree 22
Conductor 32403240
Sign 0.939−0.342i0.939 - 0.342i
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  − 2-s + 4-s + (−0.5 − 0.866i)5-s − 8-s + (0.5 + 0.866i)10-s + 16-s + 1.73i·17-s + 19-s + (−0.5 − 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (−1.5 + 0.866i)31-s − 32-s − 1.73i·34-s − 38-s + ⋯
L(s)  = 1  − 2-s + 4-s + (−0.5 − 0.866i)5-s − 8-s + (0.5 + 0.866i)10-s + 16-s + 1.73i·17-s + 19-s + (−0.5 − 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (−1.5 + 0.866i)31-s − 32-s − 1.73i·34-s − 38-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s) L(s)=((0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.939−0.342i0.939 - 0.342i
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), 0.939−0.342i)(2,\ 3240,\ (\ :0),\ 0.939 - 0.342i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.69520864310.6952086431
L(12)L(\frac12) ≈\approx 0.69520864310.6952086431
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+T 1 + T
3 1 1
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good7 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
11 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
13 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
17 1−1.73iT−T2 1 - 1.73iT - 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+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
37 1+T2 1 + T^{2}
41 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
53 1−T+T2 1 - T + T^{2}
59 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
61 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 0.866i)T + (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+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
83 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}
89 1+T2 1 + 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

−8.795877625995565746022054425171, −8.322253063691103526232715978680, −7.46180425083176276816000841178, −7.00888502137491555693564544509, −5.76722528973512968031969162656, −5.35751382373342910778605412372, −4.00060221469824650592778499589, −3.34875778136731301969430803148, −1.95272012553594927667455717285, −1.07774004749337072189208513184, 0.70540924740832342775693901353, 2.30070042155526006335616633926, 2.94601141062039916783181935240, 3.85773602517784727052389470813, 5.11847098845601104547859308480, 5.98611218435903779852507887347, 6.97886813211644625743582875593, 7.27912523533879543231904159018, 7.937226770852492521139951155570, 8.892129304618549294684432002935

Graph of the ZZ-function along the critical line