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Properties

Label 2-312-104.77-c3-0-15
Degree 22
Conductor 312312
Sign 0.535−0.844i0.535 - 0.844i
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  + (0.457 − 2.79i)2-s + 3i·3-s + (−7.58 − 2.55i)4-s − 4.56·5-s + (8.37 + 1.37i)6-s − 23.0i·7-s + (−10.5 + 19.9i)8-s − 9·9-s + (−2.08 + 12.7i)10-s + 9.02·11-s + (7.65 − 22.7i)12-s + (−3.63 + 46.7i)13-s + (−64.1 − 10.5i)14-s − 13.7i·15-s + (50.9 + 38.7i)16-s − 59.4·17-s + ⋯
L(s)  = 1  + (0.161 − 0.986i)2-s + 0.577i·3-s + (−0.947 − 0.319i)4-s − 0.408·5-s + (0.569 + 0.0933i)6-s − 1.24i·7-s + (−0.468 + 0.883i)8-s − 0.333·9-s + (−0.0660 + 0.403i)10-s + 0.247·11-s + (0.184 − 0.547i)12-s + (−0.0776 + 0.996i)13-s + (−1.22 − 0.200i)14-s − 0.235i·15-s + (0.796 + 0.604i)16-s − 0.847·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.535−0.844i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.535−0.844i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.535−0.844i0.535 - 0.844i
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), 0.535−0.844i)(2,\ 312,\ (\ :3/2),\ 0.535 - 0.844i)

Particular Values

L(2)L(2) ≈\approx 0.78846173530.7884617353
L(12)L(\frac12) ≈\approx 0.78846173530.7884617353
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+(−0.457+2.79i)T 1 + (-0.457 + 2.79i)T
3 1−3iT 1 - 3iT
13 1+(3.63−46.7i)T 1 + (3.63 - 46.7i)T
good5 1+4.56T+125T2 1 + 4.56T + 125T^{2}
7 1+23.0iT−343T2 1 + 23.0iT - 343T^{2}
11 1−9.02T+1.33e3T2 1 - 9.02T + 1.33e3T^{2}
17 1+59.4T+4.91e3T2 1 + 59.4T + 4.91e3T^{2}
19 1−27.0T+6.85e3T2 1 - 27.0T + 6.85e3T^{2}
23 1−35.7T+1.21e4T2 1 - 35.7T + 1.21e4T^{2}
29 1−142.iT−2.43e4T2 1 - 142. iT - 2.43e4T^{2}
31 1−303.iT−2.97e4T2 1 - 303. iT - 2.97e4T^{2}
37 1−241.T+5.06e4T2 1 - 241.T + 5.06e4T^{2}
41 1−280.iT−6.89e4T2 1 - 280. iT - 6.89e4T^{2}
43 1−198.iT−7.95e4T2 1 - 198. iT - 7.95e4T^{2}
47 1−319.iT−1.03e5T2 1 - 319. iT - 1.03e5T^{2}
53 1−531.iT−1.48e5T2 1 - 531. iT - 1.48e5T^{2}
59 1+378.T+2.05e5T2 1 + 378.T + 2.05e5T^{2}
61 1+878.iT−2.26e5T2 1 + 878. iT - 2.26e5T^{2}
67 1+247.T+3.00e5T2 1 + 247.T + 3.00e5T^{2}
71 1−182.iT−3.57e5T2 1 - 182. iT - 3.57e5T^{2}
73 1+815.iT−3.89e5T2 1 + 815. iT - 3.89e5T^{2}
79 1+817.T+4.93e5T2 1 + 817.T + 4.93e5T^{2}
83 1−246.T+5.71e5T2 1 - 246.T + 5.71e5T^{2}
89 1+355.iT−7.04e5T2 1 + 355. iT - 7.04e5T^{2}
97 1+506.iT−9.12e5T2 1 + 506. 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

−11.18678390501496189503485160599, −10.70378961851344839449185072317, −9.654055871443995520430688663705, −8.945505394839683755568505899850, −7.70344500720104426749594598967, −6.44282991343427040395004563343, −4.77473899967453452036849147174, −4.18733610341377570301551111224, −3.13143757850359459782754126303, −1.35130159795524443482961667975, 0.29403411211544446040597298569, 2.52109064090467456732105108831, 4.02826198240198967874517905953, 5.45332511841025867941507452630, 6.07981520439377211280172808282, 7.27345899462893927856996352862, 8.108453037323371702892475800057, 8.865296127476234615376738363802, 9.837425290656354022334400689430, 11.44178039231807409559943673376

Graph of the ZZ-function along the critical line