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Properties

Label 2-312-104.77-c3-0-12
Degree 22
Conductor 312312
Sign 0.233−0.972i0.233 - 0.972i
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  + (−2.73 − 0.722i)2-s − 3i·3-s + (6.95 + 3.95i)4-s − 2.18·5-s + (−2.16 + 8.20i)6-s − 4.02i·7-s + (−16.1 − 15.8i)8-s − 9·9-s + (5.98 + 1.58i)10-s + 8.07·11-s + (11.8 − 20.8i)12-s + (−40.2 + 24.0i)13-s + (−2.91 + 11.0i)14-s + 6.56i·15-s + (32.7 + 54.9i)16-s + 130.·17-s + ⋯
L(s)  = 1  + (−0.966 − 0.255i)2-s − 0.577i·3-s + (0.869 + 0.493i)4-s − 0.195·5-s + (−0.147 + 0.558i)6-s − 0.217i·7-s + (−0.714 − 0.699i)8-s − 0.333·9-s + (0.189 + 0.0500i)10-s + 0.221·11-s + (0.285 − 0.501i)12-s + (−0.858 + 0.513i)13-s + (−0.0555 + 0.210i)14-s + 0.113i·15-s + (0.511 + 0.858i)16-s + 1.85·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.233−0.972i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.233−0.972i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.233−0.972i0.233 - 0.972i
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), 0.233−0.972i)(2,\ 312,\ (\ :3/2),\ 0.233 - 0.972i)

Particular Values

L(2)L(2) ≈\approx 0.50577287550.5057728755
L(12)L(\frac12) ≈\approx 0.50577287550.5057728755
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+(2.73+0.722i)T 1 + (2.73 + 0.722i)T
3 1+3iT 1 + 3iT
13 1+(40.2−24.0i)T 1 + (40.2 - 24.0i)T
good5 1+2.18T+125T2 1 + 2.18T + 125T^{2}
7 1+4.02iT−343T2 1 + 4.02iT - 343T^{2}
11 1−8.07T+1.33e3T2 1 - 8.07T + 1.33e3T^{2}
17 1−130.T+4.91e3T2 1 - 130.T + 4.91e3T^{2}
19 1+109.T+6.85e3T2 1 + 109.T + 6.85e3T^{2}
23 1+77.1T+1.21e4T2 1 + 77.1T + 1.21e4T^{2}
29 1+94.2iT−2.43e4T2 1 + 94.2iT - 2.43e4T^{2}
31 1−171.iT−2.97e4T2 1 - 171. iT - 2.97e4T^{2}
37 1+18.4T+5.06e4T2 1 + 18.4T + 5.06e4T^{2}
41 1−345.iT−6.89e4T2 1 - 345. iT - 6.89e4T^{2}
43 1−272.iT−7.95e4T2 1 - 272. iT - 7.95e4T^{2}
47 1+278.iT−1.03e5T2 1 + 278. iT - 1.03e5T^{2}
53 1−443.iT−1.48e5T2 1 - 443. iT - 1.48e5T^{2}
59 1−25.5T+2.05e5T2 1 - 25.5T + 2.05e5T^{2}
61 1−606.iT−2.26e5T2 1 - 606. iT - 2.26e5T^{2}
67 1−583.T+3.00e5T2 1 - 583.T + 3.00e5T^{2}
71 1−847.iT−3.57e5T2 1 - 847. iT - 3.57e5T^{2}
73 1−1.20e3iT−3.89e5T2 1 - 1.20e3iT - 3.89e5T^{2}
79 1+1.18e3T+4.93e5T2 1 + 1.18e3T + 4.93e5T^{2}
83 1−932.T+5.71e5T2 1 - 932.T + 5.71e5T^{2}
89 1+227.iT−7.04e5T2 1 + 227. iT - 7.04e5T^{2}
97 1+710.iT−9.12e5T2 1 + 710. 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

−11.55709450742505774821753384316, −10.31310015996130299565043076899, −9.690304628267356136253143475353, −8.471667313436460523285118533573, −7.73999328625144500869022340411, −6.89716958262368666752914537289, −5.82793920420268244745871117088, −4.00805413017369439723361059240, −2.57720979129849741517076338613, −1.27753929219485460173474617584, 0.26507450132968055833835070411, 2.15605884907031145629394991764, 3.63109986874521961763994052248, 5.24577340348514537064626929094, 6.13683384574623218531761082819, 7.49593068611087545293317358230, 8.183992930571683403368302867536, 9.265295010923301934939047069178, 10.04602160904693767532863053039, 10.69383755996300042924077967097

Graph of the ZZ-function along the critical line