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Properties

Label 2-312-312.155-c1-0-30
Degree 22
Conductor 312312
Sign 0.443+0.896i0.443 + 0.896i
Analytic cond. 2.491332.49133
Root an. cond. 1.578391.57839
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−1.55 + 0.767i)3-s − 2.00·4-s − 0.380i·5-s + (−1.08 − 2.19i)6-s − 2.28·7-s − 2.82i·8-s + (1.82 − 2.38i)9-s + 0.538·10-s + (3.10 − 1.53i)12-s − 3.60·13-s − 3.23i·14-s + (0.292 + 0.591i)15-s + 4.00·16-s − 7.83i·17-s + (3.37 + 2.57i)18-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.896 + 0.443i)3-s − 1.00·4-s − 0.170i·5-s + (−0.443 − 0.896i)6-s − 0.863·7-s − 1.00i·8-s + (0.607 − 0.794i)9-s + 0.170·10-s + (0.896 − 0.443i)12-s − 1.00·13-s − 0.863i·14-s + (0.0754 + 0.152i)15-s + 1.00·16-s − 1.90i·17-s + (0.794 + 0.607i)18-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.443+0.896i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+1/2) L(s)=((0.443+0.896i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.443+0.896i0.443 + 0.896i
Analytic conductor: 2.491332.49133
Root analytic conductor: 1.578391.57839
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ312(155,⋅)\chi_{312} (155, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 312, ( :1/2), 0.443+0.896i)(2,\ 312,\ (\ :1/2),\ 0.443 + 0.896i)

Particular Values

L(1)L(1) ≈\approx 0.219701−0.136482i0.219701 - 0.136482i
L(12)L(\frac12) ≈\approx 0.219701−0.136482i0.219701 - 0.136482i
L(32)L(\frac{3}{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 1−1.41iT 1 - 1.41iT
3 1+(1.55−0.767i)T 1 + (1.55 - 0.767i)T
13 1+3.60T 1 + 3.60T
good5 1+0.380iT−5T2 1 + 0.380iT - 5T^{2}
7 1+2.28T+7T2 1 + 2.28T + 7T^{2}
11 1+11T2 1 + 11T^{2}
17 1+7.83iT−17T2 1 + 7.83iT - 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1+7.21T+31T2 1 + 7.21T + 31T^{2}
37 1+8.79T+37T2 1 + 8.79T + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+10.3T+43T2 1 + 10.3T + 43T^{2}
47 1+13.5iT−47T2 1 + 13.5iT - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1−12.7iT−71T2 1 - 12.7iT - 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1−97T2 1 - 97T^{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

−11.65133541419484359875122874800, −10.25554400946308791614960573156, −9.623345201683286626388382891229, −8.818750415132438868678351804107, −7.16899783264523922554126518871, −6.79526842487234506650208873690, −5.40694147452888918360582136569, −4.86630223078032223238333588576, −3.42488526538839223362232812270, −0.20697619039409394745641629213, 1.78968380181422057476564394585, 3.35808257439132874705872679781, 4.70490995987245699692893181151, 5.82516923682762429920774963099, 6.88107832735138830933501735548, 8.111510804251233655252644826224, 9.347540266538928316820560224185, 10.34418058825359334254738814091, 10.81103081084552777747365829438, 11.94574708215901236145182258630

Graph of the ZZ-function along the critical line