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Properties

Label 2-273-91.16-c1-0-8
Degree 22
Conductor 273273
Sign 0.731+0.681i0.731 + 0.681i
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  − 0.831·2-s + (0.5 − 0.866i)3-s − 1.30·4-s + (−1.30 + 2.26i)5-s + (−0.415 + 0.719i)6-s + (1.78 − 1.95i)7-s + 2.75·8-s + (−0.499 − 0.866i)9-s + (1.08 − 1.88i)10-s + (0.924 − 1.60i)11-s + (−0.654 + 1.13i)12-s + (2.74 − 2.33i)13-s + (−1.48 + 1.62i)14-s + (1.30 + 2.26i)15-s + 0.331·16-s + 6.83·17-s + ⋯
L(s)  = 1  − 0.587·2-s + (0.288 − 0.499i)3-s − 0.654·4-s + (−0.585 + 1.01i)5-s + (−0.169 + 0.293i)6-s + (0.673 − 0.738i)7-s + 0.972·8-s + (−0.166 − 0.288i)9-s + (0.343 − 0.595i)10-s + (0.278 − 0.482i)11-s + (−0.188 + 0.327i)12-s + (0.761 − 0.648i)13-s + (−0.396 + 0.434i)14-s + (0.337 + 0.585i)15-s + 0.0828·16-s + 1.65·17-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((0.731+0.681i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=((0.731+0.681i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 0.731+0.681i0.731 + 0.681i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(16,⋅)\chi_{273} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 0.731+0.681i)(2,\ 273,\ (\ :1/2),\ 0.731 + 0.681i)

Particular Values

L(1)L(1) ≈\approx 0.828697−0.325998i0.828697 - 0.325998i
L(12)L(\frac12) ≈\approx 0.828697−0.325998i0.828697 - 0.325998i
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+(−1.78+1.95i)T 1 + (-1.78 + 1.95i)T
13 1+(−2.74+2.33i)T 1 + (-2.74 + 2.33i)T
good2 1+0.831T+2T2 1 + 0.831T + 2T^{2}
5 1+(1.30−2.26i)T+(−2.5−4.33i)T2 1 + (1.30 - 2.26i)T + (-2.5 - 4.33i)T^{2}
11 1+(−0.924+1.60i)T+(−5.5−9.52i)T2 1 + (-0.924 + 1.60i)T + (-5.5 - 9.52i)T^{2}
17 1−6.83T+17T2 1 - 6.83T + 17T^{2}
19 1+(2.53+4.39i)T+(−9.5+16.4i)T2 1 + (2.53 + 4.39i)T + (-9.5 + 16.4i)T^{2}
23 1+1.27T+23T2 1 + 1.27T + 23T^{2}
29 1+(−0.724−1.25i)T+(−14.5+25.1i)T2 1 + (-0.724 - 1.25i)T + (-14.5 + 25.1i)T^{2}
31 1+(3.09+5.36i)T+(−15.5+26.8i)T2 1 + (3.09 + 5.36i)T + (-15.5 + 26.8i)T^{2}
37 1−7.87T+37T2 1 - 7.87T + 37T^{2}
41 1+(−4.41−7.64i)T+(−20.5+35.5i)T2 1 + (-4.41 - 7.64i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.109+0.189i)T+(−21.5−37.2i)T2 1 + (-0.109 + 0.189i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.624+1.08i)T+(−23.5−40.7i)T2 1 + (-0.624 + 1.08i)T + (-23.5 - 40.7i)T^{2}
53 1+(−1.33−2.32i)T+(−26.5+45.8i)T2 1 + (-1.33 - 2.32i)T + (-26.5 + 45.8i)T^{2}
59 1+12.0T+59T2 1 + 12.0T + 59T^{2}
61 1+(−4.36−7.55i)T+(−30.5+52.8i)T2 1 + (-4.36 - 7.55i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.91−11.9i)T+(−33.5−58.0i)T2 1 + (6.91 - 11.9i)T + (-33.5 - 58.0i)T^{2}
71 1+(−1.78+3.09i)T+(−35.5−61.4i)T2 1 + (-1.78 + 3.09i)T + (-35.5 - 61.4i)T^{2}
73 1+(3.26+5.65i)T+(−36.5+63.2i)T2 1 + (3.26 + 5.65i)T + (-36.5 + 63.2i)T^{2}
79 1+(−3.08+5.33i)T+(−39.5−68.4i)T2 1 + (-3.08 + 5.33i)T + (-39.5 - 68.4i)T^{2}
83 1+8.67T+83T2 1 + 8.67T + 83T^{2}
89 1+15.1T+89T2 1 + 15.1T + 89T^{2}
97 1+(−6.08+10.5i)T+(−48.5−84.0i)T2 1 + (-6.08 + 10.5i)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

−11.46649228659099017807377722898, −10.84744218970497641128800073550, −9.943631270564270523035448379693, −8.710883437532803834860790578013, −7.82267398268925229595116823733, −7.36341528432067934783024829182, −5.92142337250900525162389811332, −4.28382769991409376523461792515, −3.19016661966284561660177921283, −1.01038734614467746215122824393, 1.48577486305277133139762548770, 3.84591363846646054857197669544, 4.66941691218197822335558178707, 5.71107270475927664882825014889, 7.73924248527826972616332859791, 8.348544853803201447372651120576, 9.039410791129737983491409035771, 9.825692145016378008068612101296, 10.93801212619375801233885584050, 12.13765396938422315948782307399

Graph of the ZZ-function along the critical line