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Properties

Label 2-273-91.9-c1-0-16
Degree 22
Conductor 273273
Sign 0.0657+0.997i0.0657 + 0.997i
Analytic cond. 2.179912.17991
Root an. cond. 1.476451.47645
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  + (−0.328 − 0.568i)2-s + 3-s + (0.784 − 1.35i)4-s + (−0.0109 + 0.0190i)5-s + (−0.328 − 0.568i)6-s + (−1.07 − 2.41i)7-s − 2.34·8-s + 9-s + 0.0144·10-s + 3.39·11-s + (0.784 − 1.35i)12-s + (−2.82 − 2.23i)13-s + (−1.02 + 1.40i)14-s + (−0.0109 + 0.0190i)15-s + (−0.799 − 1.38i)16-s + (−1.66 + 2.88i)17-s + ⋯
L(s)  = 1  + (−0.232 − 0.402i)2-s + 0.577·3-s + (0.392 − 0.679i)4-s + (−0.00490 + 0.00850i)5-s + (−0.134 − 0.232i)6-s + (−0.405 − 0.914i)7-s − 0.828·8-s + 0.333·9-s + 0.00455·10-s + 1.02·11-s + (0.226 − 0.392i)12-s + (−0.783 − 0.621i)13-s + (−0.273 + 0.375i)14-s + (−0.00283 + 0.00490i)15-s + (−0.199 − 0.346i)16-s + (−0.403 + 0.699i)17-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((0.0657+0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0657 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=((0.0657+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0657 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 0.0657+0.997i0.0657 + 0.997i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(100,⋅)\chi_{273} (100, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 0.0657+0.997i)(2,\ 273,\ (\ :1/2),\ 0.0657 + 0.997i)

Particular Values

L(1)L(1) ≈\approx 1.02254−0.957420i1.02254 - 0.957420i
L(12)L(\frac12) ≈\approx 1.02254−0.957420i1.02254 - 0.957420i
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−T 1 - T
7 1+(1.07+2.41i)T 1 + (1.07 + 2.41i)T
13 1+(2.82+2.23i)T 1 + (2.82 + 2.23i)T
good2 1+(0.328+0.568i)T+(−1+1.73i)T2 1 + (0.328 + 0.568i)T + (-1 + 1.73i)T^{2}
5 1+(0.0109−0.0190i)T+(−2.5−4.33i)T2 1 + (0.0109 - 0.0190i)T + (-2.5 - 4.33i)T^{2}
11 1−3.39T+11T2 1 - 3.39T + 11T^{2}
17 1+(1.66−2.88i)T+(−8.5−14.7i)T2 1 + (1.66 - 2.88i)T + (-8.5 - 14.7i)T^{2}
19 1−3.33T+19T2 1 - 3.33T + 19T^{2}
23 1+(−2.75−4.76i)T+(−11.5+19.9i)T2 1 + (-2.75 - 4.76i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.52+6.10i)T+(−14.5−25.1i)T2 1 + (-3.52 + 6.10i)T + (-14.5 - 25.1i)T^{2}
31 1+(3.13+5.42i)T+(−15.5+26.8i)T2 1 + (3.13 + 5.42i)T + (-15.5 + 26.8i)T^{2}
37 1+(2.45+4.25i)T+(−18.5+32.0i)T2 1 + (2.45 + 4.25i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.27+2.20i)T+(−20.5−35.5i)T2 1 + (-1.27 + 2.20i)T + (-20.5 - 35.5i)T^{2}
43 1+(−4.49−7.77i)T+(−21.5+37.2i)T2 1 + (-4.49 - 7.77i)T + (-21.5 + 37.2i)T^{2}
47 1+(4.08−7.06i)T+(−23.5−40.7i)T2 1 + (4.08 - 7.06i)T + (-23.5 - 40.7i)T^{2}
53 1+(−5.68−9.84i)T+(−26.5+45.8i)T2 1 + (-5.68 - 9.84i)T + (-26.5 + 45.8i)T^{2}
59 1+(6.09−10.5i)T+(−29.5−51.0i)T2 1 + (6.09 - 10.5i)T + (-29.5 - 51.0i)T^{2}
61 1−10.4T+61T2 1 - 10.4T + 61T^{2}
67 1−10.7T+67T2 1 - 10.7T + 67T^{2}
71 1+(3.57+6.19i)T+(−35.5+61.4i)T2 1 + (3.57 + 6.19i)T + (-35.5 + 61.4i)T^{2}
73 1+(−0.102−0.176i)T+(−36.5+63.2i)T2 1 + (-0.102 - 0.176i)T + (-36.5 + 63.2i)T^{2}
79 1+(3.42−5.92i)T+(−39.5−68.4i)T2 1 + (3.42 - 5.92i)T + (-39.5 - 68.4i)T^{2}
83 1+14.4T+83T2 1 + 14.4T + 83T^{2}
89 1+(−2.72−4.72i)T+(−44.5+77.0i)T2 1 + (-2.72 - 4.72i)T + (-44.5 + 77.0i)T^{2}
97 1+(−4.71−8.16i)T+(−48.5+84.0i)T2 1 + (-4.71 - 8.16i)T + (-48.5 + 84.0i)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

−11.47606599915238676833464382377, −10.67651342632304877102018500234, −9.680591428215765463520514237602, −9.239746675924476485639691902680, −7.68974970307324474032657577760, −6.86280388634312577382102418743, −5.69011451117674470626262677225, −4.10354839030039153625011441211, −2.85486246108600686817853845076, −1.22123858530204146913406869796, 2.37496741727909642019694104350, 3.42602916006640846648842200365, 4.98261222402315473245330135667, 6.68179508920965538372809889532, 7.00535212688924944979770977537, 8.570556326763595010182101054460, 8.896539530667126391370219810303, 9.925683250675842544585535586665, 11.46564598861291910147470371365, 12.17147606608278945248047888917

Graph of the ZZ-function along the critical line