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Properties

Label 2-312-104.77-c3-0-18
Degree 22
Conductor 312312
Sign −0.997−0.0679i-0.997 - 0.0679i
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  + (1.61 + 2.32i)2-s − 3i·3-s + (−2.78 + 7.50i)4-s − 1.96·5-s + (6.96 − 4.84i)6-s + 7.12i·7-s + (−21.9 + 5.66i)8-s − 9·9-s + (−3.16 − 4.55i)10-s + 39.6·11-s + (22.5 + 8.34i)12-s + (−14.7 + 44.4i)13-s + (−16.5 + 11.5i)14-s + 5.88i·15-s + (−48.5 − 41.7i)16-s − 20.3·17-s + ⋯
L(s)  = 1  + (0.571 + 0.820i)2-s − 0.577i·3-s + (−0.347 + 0.937i)4-s − 0.175·5-s + (0.473 − 0.329i)6-s + 0.384i·7-s + (−0.968 + 0.250i)8-s − 0.333·9-s + (−0.100 − 0.144i)10-s + 1.08·11-s + (0.541 + 0.200i)12-s + (−0.315 + 0.948i)13-s + (−0.315 + 0.219i)14-s + 0.101i·15-s + (−0.758 − 0.651i)16-s − 0.291·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.997−0.0679i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0679i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.997−0.0679i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0679i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.997−0.0679i-0.997 - 0.0679i
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), −0.997−0.0679i)(2,\ 312,\ (\ :3/2),\ -0.997 - 0.0679i)

Particular Values

L(2)L(2) ≈\approx 1.1751064431.175106443
L(12)L(\frac12) ≈\approx 1.1751064431.175106443
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+(−1.61−2.32i)T 1 + (-1.61 - 2.32i)T
3 1+3iT 1 + 3iT
13 1+(14.7−44.4i)T 1 + (14.7 - 44.4i)T
good5 1+1.96T+125T2 1 + 1.96T + 125T^{2}
7 1−7.12iT−343T2 1 - 7.12iT - 343T^{2}
11 1−39.6T+1.33e3T2 1 - 39.6T + 1.33e3T^{2}
17 1+20.3T+4.91e3T2 1 + 20.3T + 4.91e3T^{2}
19 1+78.7T+6.85e3T2 1 + 78.7T + 6.85e3T^{2}
23 1+108.T+1.21e4T2 1 + 108.T + 1.21e4T^{2}
29 1−306.iT−2.43e4T2 1 - 306. iT - 2.43e4T^{2}
31 1+122.iT−2.97e4T2 1 + 122. iT - 2.97e4T^{2}
37 1+238.T+5.06e4T2 1 + 238.T + 5.06e4T^{2}
41 1+113.iT−6.89e4T2 1 + 113. iT - 6.89e4T^{2}
43 1−443.iT−7.95e4T2 1 - 443. iT - 7.95e4T^{2}
47 1−435.iT−1.03e5T2 1 - 435. iT - 1.03e5T^{2}
53 1+496.iT−1.48e5T2 1 + 496. iT - 1.48e5T^{2}
59 1+868.T+2.05e5T2 1 + 868.T + 2.05e5T^{2}
61 1−355.iT−2.26e5T2 1 - 355. iT - 2.26e5T^{2}
67 1−792.T+3.00e5T2 1 - 792.T + 3.00e5T^{2}
71 1−208.iT−3.57e5T2 1 - 208. iT - 3.57e5T^{2}
73 1+1.07e3iT−3.89e5T2 1 + 1.07e3iT - 3.89e5T^{2}
79 1−1.22e3T+4.93e5T2 1 - 1.22e3T + 4.93e5T^{2}
83 1−701.T+5.71e5T2 1 - 701.T + 5.71e5T^{2}
89 1+14.2iT−7.04e5T2 1 + 14.2iT - 7.04e5T^{2}
97 1+1.16e3iT−9.12e5T2 1 + 1.16e3iT - 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

−12.05802638365221263600142893425, −11.13384368619659554796744603825, −9.413195573944169466435412668532, −8.708617659263362759983871252433, −7.68590293090423684488669695757, −6.68623001322530630681395252530, −6.07383643613112503615604336790, −4.69426634241820662588612337163, −3.64471917014976640182340188897, −2.00296231675693894002412812008, 0.33717267142077076445820872170, 2.12682803628114191530352427063, 3.65618098516810368778752284135, 4.28443628418535902414698655966, 5.55347705743131396151837565521, 6.56552204624418173304374340444, 8.121812880230532038973822059267, 9.220448911541678364082195969229, 10.11716548853408313609085389302, 10.73847266888425848734763714460

Graph of the ZZ-function along the critical line