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Properties

Label 2-378-21.20-c1-0-3
Degree 22
Conductor 378378
Sign −0.327−0.944i-0.327 - 0.944i
Analytic cond. 3.018343.01834
Root an. cond. 1.737331.73733
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + 1.73·5-s + (−2.5 + 0.866i)7-s − i·8-s + 1.73i·10-s + 3i·11-s + 6.92i·13-s + (−0.866 − 2.5i)14-s + 16-s + 6.92·17-s + 3.46i·19-s − 1.73·20-s − 3·22-s − 6i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.774·5-s + (−0.944 + 0.327i)7-s − 0.353i·8-s + 0.547i·10-s + 0.904i·11-s + 1.92i·13-s + (−0.231 − 0.668i)14-s + 0.250·16-s + 1.68·17-s + 0.794i·19-s − 0.387·20-s − 0.639·22-s − 1.25i·23-s + ⋯

Functional equation

Λ(s)=(378s/2 ΓC(s) L(s)=((−0.327−0.944i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(378s/2 ΓC(s+1/2) L(s)=((−0.327−0.944i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 378378    =    2⋅33⋅72 \cdot 3^{3} \cdot 7
Sign: −0.327−0.944i-0.327 - 0.944i
Analytic conductor: 3.018343.01834
Root analytic conductor: 1.737331.73733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ378(377,⋅)\chi_{378} (377, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 378, ( :1/2), −0.327−0.944i)(2,\ 378,\ (\ :1/2),\ -0.327 - 0.944i)

Particular Values

L(1)L(1) ≈\approx 0.733330+1.03011i0.733330 + 1.03011i
L(12)L(\frac12) ≈\approx 0.733330+1.03011i0.733330 + 1.03011i
L(32)L(\frac{3}{2}) 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−iT 1 - iT
3 1 1
7 1+(2.5−0.866i)T 1 + (2.5 - 0.866i)T
good5 1−1.73T+5T2 1 - 1.73T + 5T^{2}
11 1−3iT−11T2 1 - 3iT - 11T^{2}
13 1−6.92iT−13T2 1 - 6.92iT - 13T^{2}
17 1−6.92T+17T2 1 - 6.92T + 17T^{2}
19 1−3.46iT−19T2 1 - 3.46iT - 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1−6iT−29T2 1 - 6iT - 29T^{2}
31 1+5.19iT−31T2 1 + 5.19iT - 31T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1−3.46T+41T2 1 - 3.46T + 41T^{2}
43 1+2T+43T2 1 + 2T + 43T^{2}
47 1+3.46T+47T2 1 + 3.46T + 47T^{2}
53 1+3iT−53T2 1 + 3iT - 53T^{2}
59 1−3.46T+59T2 1 - 3.46T + 59T^{2}
61 1+6.92iT−61T2 1 + 6.92iT - 61T^{2}
67 1−2T+67T2 1 - 2T + 67T^{2}
71 1+12iT−71T2 1 + 12iT - 71T^{2}
73 1+12.1iT−73T2 1 + 12.1iT - 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+1.73T+83T2 1 + 1.73T + 83T^{2}
89 1−10.3T+89T2 1 - 10.3T + 89T^{2}
97 1−12.1iT−97T2 1 - 12.1iT - 97T^{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

−11.95897178575845675128622839395, −10.33840165168140928444411217414, −9.624812580510742038863350211755, −9.090393223844280648885121274127, −7.77949847293907118173619831348, −6.70024093427145763703808662451, −6.10793565913202028806178696743, −4.96325070707529899734466334736, −3.68526917680665035907942788732, −1.97329068957981973631281959760, 0.899611126626033953425390149797, 2.86694919426937825445757505001, 3.55858479146466087117254049237, 5.42256276793357544650064973651, 5.91496784936962241399101036041, 7.44445268402087159160073099768, 8.439673463199116733588540497238, 9.710099163967345657983701077059, 10.03428170579697789839389304342, 10.94054348308868652549869382528

Graph of the ZZ-function along the critical line