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Properties

Label 2-312-104.77-c3-0-78
Degree 22
Conductor 312312
Sign −0.555−0.831i-0.555 - 0.831i
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  + (−2.23 + 1.73i)2-s − 3i·3-s + (1.96 − 7.75i)4-s − 12.2·5-s + (5.21 + 6.69i)6-s − 32.4i·7-s + (9.07 + 20.7i)8-s − 9·9-s + (27.4 − 21.3i)10-s − 68.1·11-s + (−23.2 − 5.89i)12-s + (−8.22 − 46.1i)13-s + (56.2 + 72.3i)14-s + 36.8i·15-s + (−56.2 − 30.4i)16-s + 86.0·17-s + ⋯
L(s)  = 1  + (−0.789 + 0.614i)2-s − 0.577i·3-s + (0.245 − 0.969i)4-s − 1.09·5-s + (0.354 + 0.455i)6-s − 1.74i·7-s + (0.401 + 0.915i)8-s − 0.333·9-s + (0.866 − 0.674i)10-s − 1.86·11-s + (−0.559 − 0.141i)12-s + (−0.175 − 0.984i)13-s + (1.07 + 1.38i)14-s + 0.634i·15-s + (−0.879 − 0.476i)16-s + 1.22·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.555−0.831i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.555−0.831i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.555−0.831i-0.555 - 0.831i
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), −0.555−0.831i)(2,\ 312,\ (\ :3/2),\ -0.555 - 0.831i)

Particular Values

L(2)L(2) ≈\approx 0.065743214690.06574321469
L(12)L(\frac12) ≈\approx 0.065743214690.06574321469
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+(2.23−1.73i)T 1 + (2.23 - 1.73i)T
3 1+3iT 1 + 3iT
13 1+(8.22+46.1i)T 1 + (8.22 + 46.1i)T
good5 1+12.2T+125T2 1 + 12.2T + 125T^{2}
7 1+32.4iT−343T2 1 + 32.4iT - 343T^{2}
11 1+68.1T+1.33e3T2 1 + 68.1T + 1.33e3T^{2}
17 1−86.0T+4.91e3T2 1 - 86.0T + 4.91e3T^{2}
19 1−16.5T+6.85e3T2 1 - 16.5T + 6.85e3T^{2}
23 1+41.2T+1.21e4T2 1 + 41.2T + 1.21e4T^{2}
29 1+21.6iT−2.43e4T2 1 + 21.6iT - 2.43e4T^{2}
31 1−111.iT−2.97e4T2 1 - 111. iT - 2.97e4T^{2}
37 1−410.T+5.06e4T2 1 - 410.T + 5.06e4T^{2}
41 1+253.iT−6.89e4T2 1 + 253. iT - 6.89e4T^{2}
43 1−341.iT−7.95e4T2 1 - 341. iT - 7.95e4T^{2}
47 1−360.iT−1.03e5T2 1 - 360. iT - 1.03e5T^{2}
53 1+226.iT−1.48e5T2 1 + 226. iT - 1.48e5T^{2}
59 1+725.T+2.05e5T2 1 + 725.T + 2.05e5T^{2}
61 1−118.iT−2.26e5T2 1 - 118. iT - 2.26e5T^{2}
67 1+685.T+3.00e5T2 1 + 685.T + 3.00e5T^{2}
71 1+312.iT−3.57e5T2 1 + 312. iT - 3.57e5T^{2}
73 1−736.iT−3.89e5T2 1 - 736. iT - 3.89e5T^{2}
79 1+354.T+4.93e5T2 1 + 354.T + 4.93e5T^{2}
83 1−798.T+5.71e5T2 1 - 798.T + 5.71e5T^{2}
89 1−509.iT−7.04e5T2 1 - 509. iT - 7.04e5T^{2}
97 1+1.79e3iT−9.12e5T2 1 + 1.79e3iT - 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

−10.54311927386405264345108312834, −9.865953505906182825041657121590, −8.011091639139828668724218196519, −7.81461889917859228706071153137, −7.29775127926620531975387626796, −5.84736473493118618288594655199, −4.64303544426031398575138067658, −3.05913936422437907915696866197, −0.946556906267195167939346255204, −0.03995882831199853769627765786, 2.31755955365996856523048871732, 3.24931916640187728568383820251, 4.64274634581402692862125343988, 5.84871008934797164018010797847, 7.63859502248739759243181344168, 8.138551050358704304036879985217, 9.124398264603493330820534900502, 9.926985021613636225027102808723, 10.95743604216601079222430625217, 11.86528001664524334920214749340

Graph of the ZZ-function along the critical line