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Properties

Label 2-273-91.54-c1-0-11
Degree 22
Conductor 273273
Sign 0.978−0.208i0.978 - 0.208i
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  + (0.837 − 0.837i)2-s + (0.866 + 0.5i)3-s + 0.598i·4-s + (−1.58 + 0.424i)5-s + (1.14 − 0.306i)6-s + (2.46 + 0.955i)7-s + (2.17 + 2.17i)8-s + (0.499 + 0.866i)9-s + (−0.970 + 1.68i)10-s + (−2.81 + 0.754i)11-s + (−0.299 + 0.518i)12-s + (2.68 − 2.40i)13-s + (2.86 − 1.26i)14-s + (−1.58 − 0.424i)15-s + 2.44·16-s − 0.811·17-s + ⋯
L(s)  = 1  + (0.591 − 0.591i)2-s + (0.499 + 0.288i)3-s + 0.299i·4-s + (−0.708 + 0.189i)5-s + (0.466 − 0.125i)6-s + (0.932 + 0.361i)7-s + (0.769 + 0.769i)8-s + (0.166 + 0.288i)9-s + (−0.306 + 0.531i)10-s + (−0.849 + 0.227i)11-s + (−0.0863 + 0.149i)12-s + (0.743 − 0.668i)13-s + (0.765 − 0.338i)14-s + (−0.408 − 0.109i)15-s + 0.611·16-s − 0.196·17-s + ⋯

Functional equation

Λ(s)=(273s/2 ΓC(s) L(s)=((0.978−0.208i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(273s/2 ΓC(s+1/2) L(s)=((0.978−0.208i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 273273    =    3⋅7⋅133 \cdot 7 \cdot 13
Sign: 0.978−0.208i0.978 - 0.208i
Analytic conductor: 2.179912.17991
Root analytic conductor: 1.476451.47645
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ273(145,⋅)\chi_{273} (145, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 273, ( :1/2), 0.978−0.208i)(2,\ 273,\ (\ :1/2),\ 0.978 - 0.208i)

Particular Values

L(1)L(1) ≈\approx 1.92186+0.202317i1.92186 + 0.202317i
L(12)L(\frac12) ≈\approx 1.92186+0.202317i1.92186 + 0.202317i
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
7 1+(−2.46−0.955i)T 1 + (-2.46 - 0.955i)T
13 1+(−2.68+2.40i)T 1 + (-2.68 + 2.40i)T
good2 1+(−0.837+0.837i)T−2iT2 1 + (-0.837 + 0.837i)T - 2iT^{2}
5 1+(1.58−0.424i)T+(4.33−2.5i)T2 1 + (1.58 - 0.424i)T + (4.33 - 2.5i)T^{2}
11 1+(2.81−0.754i)T+(9.52−5.5i)T2 1 + (2.81 - 0.754i)T + (9.52 - 5.5i)T^{2}
17 1+0.811T+17T2 1 + 0.811T + 17T^{2}
19 1+(−1.66+6.23i)T+(−16.4−9.5i)T2 1 + (-1.66 + 6.23i)T + (-16.4 - 9.5i)T^{2}
23 1−1.83iT−23T2 1 - 1.83iT - 23T^{2}
29 1+(2.42+4.20i)T+(−14.5+25.1i)T2 1 + (2.42 + 4.20i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.01+3.78i)T+(−26.8−15.5i)T2 1 + (-1.01 + 3.78i)T + (-26.8 - 15.5i)T^{2}
37 1+(2.89+2.89i)T+37iT2 1 + (2.89 + 2.89i)T + 37iT^{2}
41 1+(0.392−1.46i)T+(−35.5−20.5i)T2 1 + (0.392 - 1.46i)T + (-35.5 - 20.5i)T^{2}
43 1+(2.14+1.23i)T+(21.5+37.2i)T2 1 + (2.14 + 1.23i)T + (21.5 + 37.2i)T^{2}
47 1+(0.325+1.21i)T+(−40.7+23.5i)T2 1 + (0.325 + 1.21i)T + (-40.7 + 23.5i)T^{2}
53 1+(1.12+1.95i)T+(−26.5+45.8i)T2 1 + (1.12 + 1.95i)T + (-26.5 + 45.8i)T^{2}
59 1+(−4.73+4.73i)T−59iT2 1 + (-4.73 + 4.73i)T - 59iT^{2}
61 1+(0.446−0.258i)T+(30.5−52.8i)T2 1 + (0.446 - 0.258i)T + (30.5 - 52.8i)T^{2}
67 1+(−3.33−12.4i)T+(−58.0+33.5i)T2 1 + (-3.33 - 12.4i)T + (-58.0 + 33.5i)T^{2}
71 1+(2.21+8.27i)T+(−61.4+35.5i)T2 1 + (2.21 + 8.27i)T + (-61.4 + 35.5i)T^{2}
73 1+(9.87+2.64i)T+(63.2+36.5i)T2 1 + (9.87 + 2.64i)T + (63.2 + 36.5i)T^{2}
79 1+(2.17−3.76i)T+(−39.5−68.4i)T2 1 + (2.17 - 3.76i)T + (-39.5 - 68.4i)T^{2}
83 1+(7.44+7.44i)T+83iT2 1 + (7.44 + 7.44i)T + 83iT^{2}
89 1+(−5.98+5.98i)T−89iT2 1 + (-5.98 + 5.98i)T - 89iT^{2}
97 1+(−16.5+4.44i)T+(84.0−48.5i)T2 1 + (-16.5 + 4.44i)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

−11.67475148016413986501165965305, −11.32493288771572672397653529976, −10.37024780597729455274670624304, −8.905417599147295359042850617932, −8.001931692707759400370512427724, −7.44171530640393925711762604893, −5.41573101962113262548151309280, −4.46219550974320414255062518146, −3.38822832574335303092881571566, −2.27632405952145236686034220141, 1.53543114038400928372591079062, 3.68409500200751893883369678805, 4.65714092809759655907156226057, 5.77634256928465985585927103353, 7.01144884925126966429004060936, 7.88283069951824788995644896503, 8.616055969694366587842423515718, 10.09225179599906426172012704211, 10.95514990594787092951083888320, 11.99493935935898494925750850337

Graph of the ZZ-function along the critical line