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Properties

Label 2-312-104.77-c3-0-29
Degree 22
Conductor 312312
Sign −0.213−0.976i-0.213 - 0.976i
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  + (−1.33 + 2.49i)2-s − 3i·3-s + (−4.41 − 6.67i)4-s + 12.3·5-s + (7.47 + 4.01i)6-s + 26.6i·7-s + (22.5 − 2.06i)8-s − 9·9-s + (−16.5 + 30.8i)10-s − 33.2·11-s + (−20.0 + 13.2i)12-s + (46.5 − 5.78i)13-s + (−66.3 − 35.6i)14-s − 37.1i·15-s + (−25.0 + 58.8i)16-s + 16.3·17-s + ⋯
L(s)  = 1  + (−0.473 + 0.880i)2-s − 0.577i·3-s + (−0.551 − 0.834i)4-s + 1.10·5-s + (0.508 + 0.273i)6-s + 1.43i·7-s + (0.995 − 0.0911i)8-s − 0.333·9-s + (−0.523 + 0.974i)10-s − 0.911·11-s + (−0.481 + 0.318i)12-s + (0.992 − 0.123i)13-s + (−1.26 − 0.680i)14-s − 0.638i·15-s + (−0.391 + 0.920i)16-s + 0.232·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.213−0.976i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.213−0.976i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.213−0.976i-0.213 - 0.976i
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), −0.213−0.976i)(2,\ 312,\ (\ :3/2),\ -0.213 - 0.976i)

Particular Values

L(2)L(2) ≈\approx 1.4448651511.444865151
L(12)L(\frac12) ≈\approx 1.4448651511.444865151
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+(1.33−2.49i)T 1 + (1.33 - 2.49i)T
3 1+3iT 1 + 3iT
13 1+(−46.5+5.78i)T 1 + (-46.5 + 5.78i)T
good5 1−12.3T+125T2 1 - 12.3T + 125T^{2}
7 1−26.6iT−343T2 1 - 26.6iT - 343T^{2}
11 1+33.2T+1.33e3T2 1 + 33.2T + 1.33e3T^{2}
17 1−16.3T+4.91e3T2 1 - 16.3T + 4.91e3T^{2}
19 1+43.9T+6.85e3T2 1 + 43.9T + 6.85e3T^{2}
23 1−60.7T+1.21e4T2 1 - 60.7T + 1.21e4T^{2}
29 1−218.iT−2.43e4T2 1 - 218. iT - 2.43e4T^{2}
31 1−12.9iT−2.97e4T2 1 - 12.9iT - 2.97e4T^{2}
37 1−295.T+5.06e4T2 1 - 295.T + 5.06e4T^{2}
41 1−241.iT−6.89e4T2 1 - 241. iT - 6.89e4T^{2}
43 1−290.iT−7.95e4T2 1 - 290. iT - 7.95e4T^{2}
47 1−385.iT−1.03e5T2 1 - 385. iT - 1.03e5T^{2}
53 1+220.iT−1.48e5T2 1 + 220. iT - 1.48e5T^{2}
59 1−310.T+2.05e5T2 1 - 310.T + 2.05e5T^{2}
61 1−156.iT−2.26e5T2 1 - 156. iT - 2.26e5T^{2}
67 1+586.T+3.00e5T2 1 + 586.T + 3.00e5T^{2}
71 1−1.14e3iT−3.57e5T2 1 - 1.14e3iT - 3.57e5T^{2}
73 1+792.iT−3.89e5T2 1 + 792. iT - 3.89e5T^{2}
79 1+1.06e3T+4.93e5T2 1 + 1.06e3T + 4.93e5T^{2}
83 1−1.48e3T+5.71e5T2 1 - 1.48e3T + 5.71e5T^{2}
89 1−216.iT−7.04e5T2 1 - 216. iT - 7.04e5T^{2}
97 1−789.iT−9.12e5T2 1 - 789. 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

−11.36429788779593588271429192996, −10.36491626838017630915639789110, −9.318160004855737600442749965093, −8.661975184091095532975861290924, −7.78219507055084595203912404113, −6.39697787492149122262957754639, −5.86940775746214712961721036347, −5.04556883808092908467625052267, −2.66379590297954384133863008641, −1.40580150503090330471506532418, 0.66204714787393980638487822392, 2.16319334727392126300187008625, 3.57589525370663841855502259461, 4.56100936331362240759740445698, 5.90037802704395272039271894943, 7.33276290001017693370602897220, 8.388493182348829083579717730408, 9.421922665909396306444933004387, 10.29682239921751268249073651615, 10.58549328893972429471539680105

Graph of the ZZ-function along the critical line