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Properties

Label 2-675-27.16-c1-0-34
Degree 22
Conductor 675675
Sign 0.950−0.311i0.950 - 0.311i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
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  + (0.672 + 0.564i)2-s + (1.45 + 0.939i)3-s + (−0.213 − 1.21i)4-s + (0.448 + 1.45i)6-s + (−0.0883 + 0.501i)7-s + (1.41 − 2.45i)8-s + (1.23 + 2.73i)9-s + (2.28 − 0.830i)11-s + (0.826 − 1.96i)12-s + (4.85 − 4.07i)13-s + (−0.342 + 0.287i)14-s + (0.0284 − 0.0103i)16-s + (−2.86 − 4.96i)17-s + (−0.711 + 2.53i)18-s + (−1.94 + 3.36i)19-s + ⋯
L(s)  = 1  + (0.475 + 0.399i)2-s + (0.840 + 0.542i)3-s + (−0.106 − 0.605i)4-s + (0.183 + 0.593i)6-s + (−0.0334 + 0.189i)7-s + (0.501 − 0.868i)8-s + (0.411 + 0.911i)9-s + (0.687 − 0.250i)11-s + (0.238 − 0.566i)12-s + (1.34 − 1.13i)13-s + (−0.0914 + 0.0767i)14-s + (0.00710 − 0.00258i)16-s + (−0.695 − 1.20i)17-s + (−0.167 + 0.597i)18-s + (−0.446 + 0.772i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.950−0.311i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((0.950−0.311i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.950−0.311i0.950 - 0.311i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(151,⋅)\chi_{675} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 0.950−0.311i)(2,\ 675,\ (\ :1/2),\ 0.950 - 0.311i)

Particular Values

L(1)L(1) ≈\approx 2.59970+0.415257i2.59970 + 0.415257i
L(12)L(\frac12) ≈\approx 2.59970+0.415257i2.59970 + 0.415257i
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+(−1.45−0.939i)T 1 + (-1.45 - 0.939i)T
5 1 1
good2 1+(−0.672−0.564i)T+(0.347+1.96i)T2 1 + (-0.672 - 0.564i)T + (0.347 + 1.96i)T^{2}
7 1+(0.0883−0.501i)T+(−6.57−2.39i)T2 1 + (0.0883 - 0.501i)T + (-6.57 - 2.39i)T^{2}
11 1+(−2.28+0.830i)T+(8.42−7.07i)T2 1 + (-2.28 + 0.830i)T + (8.42 - 7.07i)T^{2}
13 1+(−4.85+4.07i)T+(2.25−12.8i)T2 1 + (-4.85 + 4.07i)T + (2.25 - 12.8i)T^{2}
17 1+(2.86+4.96i)T+(−8.5+14.7i)T2 1 + (2.86 + 4.96i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.94−3.36i)T+(−9.5−16.4i)T2 1 + (1.94 - 3.36i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.12−6.37i)T+(−21.6+7.86i)T2 1 + (-1.12 - 6.37i)T + (-21.6 + 7.86i)T^{2}
29 1+(0.324+0.272i)T+(5.03+28.5i)T2 1 + (0.324 + 0.272i)T + (5.03 + 28.5i)T^{2}
31 1+(−1.47−8.37i)T+(−29.1+10.6i)T2 1 + (-1.47 - 8.37i)T + (-29.1 + 10.6i)T^{2}
37 1+(1.19+2.06i)T+(−18.5+32.0i)T2 1 + (1.19 + 2.06i)T + (-18.5 + 32.0i)T^{2}
41 1+(−0.938+0.787i)T+(7.11−40.3i)T2 1 + (-0.938 + 0.787i)T + (7.11 - 40.3i)T^{2}
43 1+(−2.34+0.855i)T+(32.9−27.6i)T2 1 + (-2.34 + 0.855i)T + (32.9 - 27.6i)T^{2}
47 1+(−0.143+0.816i)T+(−44.1−16.0i)T2 1 + (-0.143 + 0.816i)T + (-44.1 - 16.0i)T^{2}
53 1+8.88T+53T2 1 + 8.88T + 53T^{2}
59 1+(6.57+2.39i)T+(45.1+37.9i)T2 1 + (6.57 + 2.39i)T + (45.1 + 37.9i)T^{2}
61 1+(−0.558+3.16i)T+(−57.3−20.8i)T2 1 + (-0.558 + 3.16i)T + (-57.3 - 20.8i)T^{2}
67 1+(5.32−4.47i)T+(11.6−65.9i)T2 1 + (5.32 - 4.47i)T + (11.6 - 65.9i)T^{2}
71 1+(1.59+2.77i)T+(−35.5+61.4i)T2 1 + (1.59 + 2.77i)T + (-35.5 + 61.4i)T^{2}
73 1+(−5.99+10.3i)T+(−36.5−63.2i)T2 1 + (-5.99 + 10.3i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.06−1.73i)T+(13.7+77.7i)T2 1 + (-2.06 - 1.73i)T + (13.7 + 77.7i)T^{2}
83 1+(6.00+5.03i)T+(14.4+81.7i)T2 1 + (6.00 + 5.03i)T + (14.4 + 81.7i)T^{2}
89 1+(9.24−16.0i)T+(−44.5−77.0i)T2 1 + (9.24 - 16.0i)T + (-44.5 - 77.0i)T^{2}
97 1+(−7.86+2.86i)T+(74.3−62.3i)T2 1 + (-7.86 + 2.86i)T + (74.3 - 62.3i)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

−10.56324322758958343187532078861, −9.507740200810321280725437374668, −8.958598891010619217956410346956, −7.991401316640782146592650423981, −6.93133253258095848147000550859, −5.90262145227319762701542033747, −5.09218112897177596797755864837, −4.00722134489322773187217705834, −3.15302100476192062332406563698, −1.43799387542788385762720925307, 1.64972470303690108350324233630, 2.69185509699818731158871774825, 4.07831915235876894041122790245, 4.21872732713367834410619839427, 6.27373570742840591765782507147, 6.84790541669992122035163290758, 8.027971524638299229215977056485, 8.703198948520899212947848636024, 9.259839744208550793782136437631, 10.71010424055825723634660538958

Graph of the ZZ-function along the critical line