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Properties

Label 2-546-91.76-c1-0-9
Degree 22
Conductor 546546
Sign 0.257+0.966i0.257 + 0.966i
Analytic cond. 4.359834.35983
Root an. cond. 2.088022.08802
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  + (0.258 − 0.965i)2-s + (−0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (−2.63 + 2.63i)5-s + (0.258 + 0.965i)6-s + (−1.51 − 2.16i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.86 + 3.22i)10-s + (4.21 + 1.12i)11-s + 12-s + (3.32 − 1.39i)13-s + (−2.48 + 0.903i)14-s + (0.963 − 3.59i)15-s + (0.500 + 0.866i)16-s + (3.14 − 5.44i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 − 0.249i)4-s + (−1.17 + 1.17i)5-s + (0.105 + 0.394i)6-s + (−0.572 − 0.819i)7-s + (−0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (0.588 + 1.01i)10-s + (1.27 + 0.340i)11-s + 0.288·12-s + (0.922 − 0.387i)13-s + (−0.664 + 0.241i)14-s + (0.248 − 0.928i)15-s + (0.125 + 0.216i)16-s + (0.762 − 1.32i)17-s + ⋯

Functional equation

Λ(s)=(546s/2 ΓC(s) L(s)=((0.257+0.966i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(546s/2 ΓC(s+1/2) L(s)=((0.257+0.966i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 546546    =    2⋅3⋅7⋅132 \cdot 3 \cdot 7 \cdot 13
Sign: 0.257+0.966i0.257 + 0.966i
Analytic conductor: 4.359834.35983
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ546(349,⋅)\chi_{546} (349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 546, ( :1/2), 0.257+0.966i)(2,\ 546,\ (\ :1/2),\ 0.257 + 0.966i)

Particular Values

L(1)L(1) ≈\approx 0.753156−0.578458i0.753156 - 0.578458i
L(12)L(\frac12) ≈\approx 0.753156−0.578458i0.753156 - 0.578458i
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+(−0.258+0.965i)T 1 + (-0.258 + 0.965i)T
3 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
7 1+(1.51+2.16i)T 1 + (1.51 + 2.16i)T
13 1+(−3.32+1.39i)T 1 + (-3.32 + 1.39i)T
good5 1+(2.63−2.63i)T−5iT2 1 + (2.63 - 2.63i)T - 5iT^{2}
11 1+(−4.21−1.12i)T+(9.52+5.5i)T2 1 + (-4.21 - 1.12i)T + (9.52 + 5.5i)T^{2}
17 1+(−3.14+5.44i)T+(−8.5−14.7i)T2 1 + (-3.14 + 5.44i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.27+4.76i)T+(−16.4+9.5i)T2 1 + (1.27 + 4.76i)T + (-16.4 + 9.5i)T^{2}
23 1+(2.06−1.19i)T+(11.5−19.9i)T2 1 + (2.06 - 1.19i)T + (11.5 - 19.9i)T^{2}
29 1+(0.949+1.64i)T+(−14.5+25.1i)T2 1 + (0.949 + 1.64i)T + (-14.5 + 25.1i)T^{2}
31 1+(−0.189+0.189i)T−31iT2 1 + (-0.189 + 0.189i)T - 31iT^{2}
37 1+(−11.2−3.01i)T+(32.0+18.5i)T2 1 + (-11.2 - 3.01i)T + (32.0 + 18.5i)T^{2}
41 1+(−4.94−1.32i)T+(35.5+20.5i)T2 1 + (-4.94 - 1.32i)T + (35.5 + 20.5i)T^{2}
43 1+(6.21+3.58i)T+(21.5+37.2i)T2 1 + (6.21 + 3.58i)T + (21.5 + 37.2i)T^{2}
47 1+(−1.20−1.20i)T+47iT2 1 + (-1.20 - 1.20i)T + 47iT^{2}
53 1−10.0T+53T2 1 - 10.0T + 53T^{2}
59 1+(−9.27+2.48i)T+(51.0−29.5i)T2 1 + (-9.27 + 2.48i)T + (51.0 - 29.5i)T^{2}
61 1+(10.6+6.17i)T+(30.5+52.8i)T2 1 + (10.6 + 6.17i)T + (30.5 + 52.8i)T^{2}
67 1+(−3.16+11.8i)T+(−58.0−33.5i)T2 1 + (-3.16 + 11.8i)T + (-58.0 - 33.5i)T^{2}
71 1+(7.71−2.06i)T+(61.4−35.5i)T2 1 + (7.71 - 2.06i)T + (61.4 - 35.5i)T^{2}
73 1+(6.68+6.68i)T+73iT2 1 + (6.68 + 6.68i)T + 73iT^{2}
79 1+11.0T+79T2 1 + 11.0T + 79T^{2}
83 1+(−4.21+4.21i)T−83iT2 1 + (-4.21 + 4.21i)T - 83iT^{2}
89 1+(1.66−6.20i)T+(−77.0−44.5i)T2 1 + (1.66 - 6.20i)T + (-77.0 - 44.5i)T^{2}
97 1+(−0.409−1.52i)T+(−84.0+48.5i)T2 1 + (-0.409 - 1.52i)T + (-84.0 + 48.5i)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

−10.79962683549546878150271711893, −10.00638268167603032582473939549, −9.186580052891900515247387355479, −7.75109380476531172517453627751, −6.93745177464254969896036797956, −6.15170829399854312012113035225, −4.51647812682213750833591693502, −3.77261840534375560917491166034, −2.99999041049097704598742854719, −0.68611050954939521461693380092, 1.22542168352905879633969639357, 3.73435727041820839557799715978, 4.26000745018103231836451704266, 5.80828771958352231246760110551, 6.10104157905661702722048817139, 7.41008483710006734721240236073, 8.542621943766790547572548550396, 8.662996775224917027064434294374, 9.934351282937442670193661075727, 11.39147455130831797496401670635

Graph of the ZZ-function along the critical line