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Properties

Label 2-312-104.77-c3-0-19
Degree 22
Conductor 312312
Sign 0.949+0.312i0.949 + 0.312i
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  + (−2.47 − 1.37i)2-s − 3i·3-s + (4.22 + 6.79i)4-s − 19.0·5-s + (−4.11 + 7.41i)6-s + 2.83i·7-s + (−1.13 − 22.5i)8-s − 9·9-s + (47.0 + 26.1i)10-s − 28.0·11-s + (20.3 − 12.6i)12-s + (−43.7 − 16.8i)13-s + (3.88 − 7.00i)14-s + 57.1i·15-s + (−28.2 + 57.4i)16-s − 86.5·17-s + ⋯
L(s)  = 1  + (−0.874 − 0.485i)2-s − 0.577i·3-s + (0.528 + 0.848i)4-s − 1.70·5-s + (−0.280 + 0.504i)6-s + 0.152i·7-s + (−0.0500 − 0.998i)8-s − 0.333·9-s + (1.48 + 0.826i)10-s − 0.769·11-s + (0.490 − 0.305i)12-s + (−0.932 − 0.359i)13-s + (0.0742 − 0.133i)14-s + 0.983i·15-s + (−0.441 + 0.897i)16-s − 1.23·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.949+0.312i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.949+0.312i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.949+0.312i0.949 + 0.312i
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), 0.949+0.312i)(2,\ 312,\ (\ :3/2),\ 0.949 + 0.312i)

Particular Values

L(2)L(2) ≈\approx 0.45439351930.4543935193
L(12)L(\frac12) ≈\approx 0.45439351930.4543935193
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+(2.47+1.37i)T 1 + (2.47 + 1.37i)T
3 1+3iT 1 + 3iT
13 1+(43.7+16.8i)T 1 + (43.7 + 16.8i)T
good5 1+19.0T+125T2 1 + 19.0T + 125T^{2}
7 1−2.83iT−343T2 1 - 2.83iT - 343T^{2}
11 1+28.0T+1.33e3T2 1 + 28.0T + 1.33e3T^{2}
17 1+86.5T+4.91e3T2 1 + 86.5T + 4.91e3T^{2}
19 1−82.3T+6.85e3T2 1 - 82.3T + 6.85e3T^{2}
23 1−81.3T+1.21e4T2 1 - 81.3T + 1.21e4T^{2}
29 1−70.2iT−2.43e4T2 1 - 70.2iT - 2.43e4T^{2}
31 1−184.iT−2.97e4T2 1 - 184. iT - 2.97e4T^{2}
37 1+62.8T+5.06e4T2 1 + 62.8T + 5.06e4T^{2}
41 1+405.iT−6.89e4T2 1 + 405. iT - 6.89e4T^{2}
43 1+257.iT−7.95e4T2 1 + 257. iT - 7.95e4T^{2}
47 1−200.iT−1.03e5T2 1 - 200. iT - 1.03e5T^{2}
53 1−121.iT−1.48e5T2 1 - 121. iT - 1.48e5T^{2}
59 1−513.T+2.05e5T2 1 - 513.T + 2.05e5T^{2}
61 1−54.0iT−2.26e5T2 1 - 54.0iT - 2.26e5T^{2}
67 1−438.T+3.00e5T2 1 - 438.T + 3.00e5T^{2}
71 1+181.iT−3.57e5T2 1 + 181. iT - 3.57e5T^{2}
73 1+922.iT−3.89e5T2 1 + 922. iT - 3.89e5T^{2}
79 1−158.T+4.93e5T2 1 - 158.T + 4.93e5T^{2}
83 1−324.T+5.71e5T2 1 - 324.T + 5.71e5T^{2}
89 1−990.iT−7.04e5T2 1 - 990. iT - 7.04e5T^{2}
97 1−1.57e3iT−9.12e5T2 1 - 1.57e3iT - 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

−11.15644847177377622120344243000, −10.53315690557968710730775270464, −9.081477955479122869362563738324, −8.328150178324247334644113117409, −7.40531482702291046983105524131, −7.01603065182728898890187761166, −5.01502258383827764468976900378, −3.57628785435341489485905052726, −2.49920983894444725647533911904, −0.62025494560991079958149125553, 0.41139479749537998394453127304, 2.72586054016691957485573958994, 4.27137747027636784071014119426, 5.18942918397386235296000839859, 6.81411218702690322558261216671, 7.60938292071686077230033248190, 8.326093203131992460041724513451, 9.339849943199741063970762510381, 10.26294596001922415300393272690, 11.39951980176477108833604768396

Graph of the ZZ-function along the critical line