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Properties

Label 2-312-104.77-c3-0-11
Degree 22
Conductor 312312
Sign 0.0324+0.999i0.0324 + 0.999i
Analytic cond. 18.408518.4085
Root an. cond. 4.290524.29052
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.33 + 2.49i)2-s + 3i·3-s + (−4.41 + 6.67i)4-s − 12.3·5-s + (−7.47 + 4.01i)6-s + 26.6i·7-s + (−22.5 − 2.06i)8-s − 9·9-s + (−16.5 − 30.8i)10-s + 33.2·11-s + (−20.0 − 13.2i)12-s + (−46.5 + 5.78i)13-s + (−66.3 + 35.6i)14-s − 37.1i·15-s + (−25.0 − 58.8i)16-s + 16.3·17-s + ⋯
L(s)  = 1  + (0.473 + 0.880i)2-s + 0.577i·3-s + (−0.551 + 0.834i)4-s − 1.10·5-s + (−0.508 + 0.273i)6-s + 1.43i·7-s + (−0.995 − 0.0911i)8-s − 0.333·9-s + (−0.523 − 0.974i)10-s + 0.911·11-s + (−0.481 − 0.318i)12-s + (−0.992 + 0.123i)13-s + (−1.26 + 0.680i)14-s − 0.638i·15-s + (−0.391 − 0.920i)16-s + 0.232·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.0324+0.999i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0324 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.0324+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0324 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.0324+0.999i0.0324 + 0.999i
Analytic conductor: 18.408518.4085
Root analytic conductor: 4.290524.29052
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ312(181,⋅)\chi_{312} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 312, ( :3/2), 0.0324+0.999i)(2,\ 312,\ (\ :3/2),\ 0.0324 + 0.999i)

Particular Values

L(2)L(2) ≈\approx 0.68915991190.6891599119
L(12)L(\frac12) ≈\approx 0.68915991190.6891599119
L(52)L(\frac{5}{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)
bad2 1+(−1.33−2.49i)T 1 + (-1.33 - 2.49i)T
3 1−3iT 1 - 3iT
13 1+(46.5−5.78i)T 1 + (46.5 - 5.78i)T
good5 1+12.3T+125T2 1 + 12.3T + 125T^{2}
7 1−26.6iT−343T2 1 - 26.6iT - 343T^{2}
11 1−33.2T+1.33e3T2 1 - 33.2T + 1.33e3T^{2}
17 1−16.3T+4.91e3T2 1 - 16.3T + 4.91e3T^{2}
19 1−43.9T+6.85e3T2 1 - 43.9T + 6.85e3T^{2}
23 1−60.7T+1.21e4T2 1 - 60.7T + 1.21e4T^{2}
29 1+218.iT−2.43e4T2 1 + 218. iT - 2.43e4T^{2}
31 1−12.9iT−2.97e4T2 1 - 12.9iT - 2.97e4T^{2}
37 1+295.T+5.06e4T2 1 + 295.T + 5.06e4T^{2}
41 1−241.iT−6.89e4T2 1 - 241. iT - 6.89e4T^{2}
43 1+290.iT−7.95e4T2 1 + 290. iT - 7.95e4T^{2}
47 1−385.iT−1.03e5T2 1 - 385. iT - 1.03e5T^{2}
53 1−220.iT−1.48e5T2 1 - 220. iT - 1.48e5T^{2}
59 1+310.T+2.05e5T2 1 + 310.T + 2.05e5T^{2}
61 1+156.iT−2.26e5T2 1 + 156. iT - 2.26e5T^{2}
67 1−586.T+3.00e5T2 1 - 586.T + 3.00e5T^{2}
71 1−1.14e3iT−3.57e5T2 1 - 1.14e3iT - 3.57e5T^{2}
73 1+792.iT−3.89e5T2 1 + 792. iT - 3.89e5T^{2}
79 1+1.06e3T+4.93e5T2 1 + 1.06e3T + 4.93e5T^{2}
83 1+1.48e3T+5.71e5T2 1 + 1.48e3T + 5.71e5T^{2}
89 1−216.iT−7.04e5T2 1 - 216. iT - 7.04e5T^{2}
97 1−789.iT−9.12e5T2 1 - 789. iT - 9.12e5T^{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

−11.92328611804066835097325241459, −11.51700674808193993123913002935, −9.715523044799194585388695253270, −8.964306702231399949270750136261, −8.122631613894856443413282903959, −7.13353500319341920379108773855, −5.93343731261589068181145456067, −4.96320845965010974710007167497, −3.98368857456383113234570758515, −2.83786875703432138912879938984, 0.22992649872063810860536264111, 1.38493062807427659442948059816, 3.24263538582534763004272469020, 4.05736577885475353062143549469, 5.14914984704579331673185596259, 6.82813291635321989146808063840, 7.44078326485470813252915145961, 8.692346309048661151341420551120, 9.874702260463115788718371121181, 10.78811692370588538774984214877

Graph of the ZZ-function along the critical line