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Properties

Label 2-1680-105.104-c1-0-46
Degree 22
Conductor 16801680
Sign 0.645−0.763i0.645 - 0.763i
Analytic cond. 13.414813.4148
Root an. cond. 3.662633.66263
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  + (1.32 + 1.11i)3-s − 2.23i·5-s + 2.64·7-s + (0.5 + 2.95i)9-s + 5.91i·11-s + 2.64·13-s + (2.50 − 2.95i)15-s + 2.23i·17-s + (3.50 + 2.95i)21-s − 5.00·25-s + (−2.64 + 4.47i)27-s − 5.91i·29-s + (−6.61 + 7.82i)33-s − 5.91i·35-s + (3.50 + 2.95i)39-s + ⋯
L(s)  = 1  + (0.763 + 0.645i)3-s − 0.999i·5-s + 0.999·7-s + (0.166 + 0.986i)9-s + 1.78i·11-s + 0.733·13-s + (0.645 − 0.763i)15-s + 0.542i·17-s + (0.763 + 0.645i)21-s − 1.00·25-s + (−0.509 + 0.860i)27-s − 1.09i·29-s + (−1.15 + 1.36i)33-s − 0.999i·35-s + (0.560 + 0.473i)39-s + ⋯

Functional equation

Λ(s)=(1680s/2 ΓC(s) L(s)=((0.645−0.763i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1680s/2 ΓC(s+1/2) L(s)=((0.645−0.763i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16801680    =    24⋅3⋅5⋅72^{4} \cdot 3 \cdot 5 \cdot 7
Sign: 0.645−0.763i0.645 - 0.763i
Analytic conductor: 13.414813.4148
Root analytic conductor: 3.662633.66263
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1680(209,⋅)\chi_{1680} (209, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1680, ( :1/2), 0.645−0.763i)(2,\ 1680,\ (\ :1/2),\ 0.645 - 0.763i)

Particular Values

L(1)L(1) ≈\approx 2.5758674972.575867497
L(12)L(\frac12) ≈\approx 2.5758674972.575867497
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 1
3 1+(−1.32−1.11i)T 1 + (-1.32 - 1.11i)T
5 1+2.23iT 1 + 2.23iT
7 1−2.64T 1 - 2.64T
good11 1−5.91iT−11T2 1 - 5.91iT - 11T^{2}
13 1−2.64T+13T2 1 - 2.64T + 13T^{2}
17 1−2.23iT−17T2 1 - 2.23iT - 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+5.91iT−29T2 1 + 5.91iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1−11.1iT−47T2 1 - 11.1iT - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+11.8iT−71T2 1 + 11.8iT - 71T^{2}
73 1−10.5T+73T2 1 - 10.5T + 73T^{2}
79 1−T+79T2 1 - T + 79T^{2}
83 1+8.94iT−83T2 1 + 8.94iT - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1−18.5T+97T2 1 - 18.5T + 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

−9.379848192735294361021552518228, −8.720386269714447252084914173897, −7.939351269325077570501980897889, −7.51081141673473194106519147718, −6.09812943164351163838147747091, −4.97478162678790030623157479396, −4.52897198952366180722656194167, −3.81058416471280730985409857525, −2.25620305537984466720236272901, −1.50504930325271593128499519220, 0.987091973113866678190838040365, 2.22092022367522043124687045394, 3.21284286801102683484996634165, 3.81300495629830335409044942467, 5.31712709551524249238507716730, 6.17222088708926379418825205322, 6.93633576982369294721554630260, 7.71744351816188103250288745195, 8.478320022035703595871627067991, 8.875575741355872677074686417395

Graph of the ZZ-function along the critical line