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Properties

Label 2-891-11.3-c1-0-1
Degree 22
Conductor 891891
Sign −0.999+0.0252i-0.999 + 0.0252i
Analytic cond. 7.114677.11467
Root an. cond. 2.667332.66733
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  + (0.537 − 0.390i)2-s + (−0.481 + 1.48i)4-s + (−0.903 − 0.656i)5-s + (−1.20 + 3.71i)7-s + (0.730 + 2.24i)8-s − 0.741·10-s + (−1.17 − 3.10i)11-s + (−2.16 + 1.57i)13-s + (0.801 + 2.46i)14-s + (−1.25 − 0.909i)16-s + (−4.48 − 3.26i)17-s + (−1.58 − 4.88i)19-s + (1.40 − 1.02i)20-s + (−1.84 − 1.20i)22-s + 2.11·23-s + ⋯
L(s)  = 1  + (0.380 − 0.276i)2-s + (−0.240 + 0.741i)4-s + (−0.403 − 0.293i)5-s + (−0.456 + 1.40i)7-s + (0.258 + 0.794i)8-s − 0.234·10-s + (−0.355 − 0.934i)11-s + (−0.600 + 0.436i)13-s + (0.214 + 0.659i)14-s + (−0.312 − 0.227i)16-s + (−1.08 − 0.790i)17-s + (−0.363 − 1.11i)19-s + (0.314 − 0.228i)20-s + (−0.393 − 0.257i)22-s + 0.440·23-s + ⋯

Functional equation

Λ(s)=(891s/2 ΓC(s) L(s)=((−0.999+0.0252i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0252i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(891s/2 ΓC(s+1/2) L(s)=((−0.999+0.0252i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0252i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 891891    =    34⋅113^{4} \cdot 11
Sign: −0.999+0.0252i-0.999 + 0.0252i
Analytic conductor: 7.114677.11467
Root analytic conductor: 2.667332.66733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ891(487,⋅)\chi_{891} (487, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 891, ( :1/2), −0.999+0.0252i)(2,\ 891,\ (\ :1/2),\ -0.999 + 0.0252i)

Particular Values

L(1)L(1) ≈\approx 0.00313446−0.248491i0.00313446 - 0.248491i
L(12)L(\frac12) ≈\approx 0.00313446−0.248491i0.00313446 - 0.248491i
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)
bad3 1 1
11 1+(1.17+3.10i)T 1 + (1.17 + 3.10i)T
good2 1+(−0.537+0.390i)T+(0.618−1.90i)T2 1 + (-0.537 + 0.390i)T + (0.618 - 1.90i)T^{2}
5 1+(0.903+0.656i)T+(1.54+4.75i)T2 1 + (0.903 + 0.656i)T + (1.54 + 4.75i)T^{2}
7 1+(1.20−3.71i)T+(−5.66−4.11i)T2 1 + (1.20 - 3.71i)T + (-5.66 - 4.11i)T^{2}
13 1+(2.16−1.57i)T+(4.01−12.3i)T2 1 + (2.16 - 1.57i)T + (4.01 - 12.3i)T^{2}
17 1+(4.48+3.26i)T+(5.25+16.1i)T2 1 + (4.48 + 3.26i)T + (5.25 + 16.1i)T^{2}
19 1+(1.58+4.88i)T+(−15.3+11.1i)T2 1 + (1.58 + 4.88i)T + (-15.3 + 11.1i)T^{2}
23 1−2.11T+23T2 1 - 2.11T + 23T^{2}
29 1+(−0.334+1.03i)T+(−23.4−17.0i)T2 1 + (-0.334 + 1.03i)T + (-23.4 - 17.0i)T^{2}
31 1+(−0.348+0.252i)T+(9.57−29.4i)T2 1 + (-0.348 + 0.252i)T + (9.57 - 29.4i)T^{2}
37 1+(2.69−8.27i)T+(−29.9−21.7i)T2 1 + (2.69 - 8.27i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.902+2.77i)T+(−33.1+24.0i)T2 1 + (0.902 + 2.77i)T + (-33.1 + 24.0i)T^{2}
43 1+2.23T+43T2 1 + 2.23T + 43T^{2}
47 1+(−3.96−12.2i)T+(−38.0+27.6i)T2 1 + (-3.96 - 12.2i)T + (-38.0 + 27.6i)T^{2}
53 1+(4.56−3.31i)T+(16.3−50.4i)T2 1 + (4.56 - 3.31i)T + (16.3 - 50.4i)T^{2}
59 1+(0.734−2.25i)T+(−47.7−34.6i)T2 1 + (0.734 - 2.25i)T + (-47.7 - 34.6i)T^{2}
61 1+(−3.30−2.40i)T+(18.8+58.0i)T2 1 + (-3.30 - 2.40i)T + (18.8 + 58.0i)T^{2}
67 1+9.75T+67T2 1 + 9.75T + 67T^{2}
71 1+(5.11+3.71i)T+(21.9+67.5i)T2 1 + (5.11 + 3.71i)T + (21.9 + 67.5i)T^{2}
73 1+(3.60−11.1i)T+(−59.0−42.9i)T2 1 + (3.60 - 11.1i)T + (-59.0 - 42.9i)T^{2}
79 1+(3.65−2.65i)T+(24.4−75.1i)T2 1 + (3.65 - 2.65i)T + (24.4 - 75.1i)T^{2}
83 1+(0.389+0.283i)T+(25.6+78.9i)T2 1 + (0.389 + 0.283i)T + (25.6 + 78.9i)T^{2}
89 1−16.0T+89T2 1 - 16.0T + 89T^{2}
97 1+(6.48−4.71i)T+(29.9−92.2i)T2 1 + (6.48 - 4.71i)T + (29.9 - 92.2i)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

−10.81674438032469175153339734650, −9.385035461703761197145522014911, −8.844619536392860011090775066152, −8.270658898310627675558199566861, −7.16429977433340249764266014758, −6.15598538229805280981916041099, −5.02543262283195631696436358317, −4.39213979010353410280182854374, −2.96181729076216591248220582151, −2.49944841795214544982682157315, 0.098975345366517127934781576575, 1.80777747624913365497027391451, 3.59460254458945808487480803691, 4.27951288744508518598733723685, 5.19740077419748683883275310594, 6.33600594442951836022897862489, 7.09850613312626439357945598662, 7.65902255220121062576356415579, 8.954919326748970610724281470052, 10.05498779233529035518455525552

Graph of the ZZ-function along the critical line