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Properties

Label 2-312-104.77-c3-0-21
Degree 22
Conductor 312312
Sign 0.0418−0.999i0.0418 - 0.999i
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  + (−0.800 − 2.71i)2-s + 3i·3-s + (−6.71 + 4.34i)4-s + 12.1·5-s + (8.13 − 2.40i)6-s + 21.1i·7-s + (17.1 + 14.7i)8-s − 9·9-s + (−9.70 − 32.8i)10-s + 16.0·11-s + (−13.0 − 20.1i)12-s + (−36.8 + 29.0i)13-s + (57.3 − 16.9i)14-s + 36.3i·15-s + (26.2 − 58.3i)16-s + 45.8·17-s + ⋯
L(s)  = 1  + (−0.283 − 0.959i)2-s + 0.577i·3-s + (−0.839 + 0.543i)4-s + 1.08·5-s + (0.553 − 0.163i)6-s + 1.14i·7-s + (0.758 + 0.651i)8-s − 0.333·9-s + (−0.307 − 1.04i)10-s + 0.438·11-s + (−0.313 − 0.484i)12-s + (−0.785 + 0.619i)13-s + (1.09 − 0.323i)14-s + 0.626i·15-s + (0.410 − 0.911i)16-s + 0.653·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.0418−0.999i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0418 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.0418−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0418 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.0418−0.999i0.0418 - 0.999i
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), 0.0418−0.999i)(2,\ 312,\ (\ :3/2),\ 0.0418 - 0.999i)

Particular Values

L(2)L(2) ≈\approx 1.1790593891.179059389
L(12)L(\frac12) ≈\approx 1.1790593891.179059389
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+(0.800+2.71i)T 1 + (0.800 + 2.71i)T
3 1−3iT 1 - 3iT
13 1+(36.8−29.0i)T 1 + (36.8 - 29.0i)T
good5 1−12.1T+125T2 1 - 12.1T + 125T^{2}
7 1−21.1iT−343T2 1 - 21.1iT - 343T^{2}
11 1−16.0T+1.33e3T2 1 - 16.0T + 1.33e3T^{2}
17 1−45.8T+4.91e3T2 1 - 45.8T + 4.91e3T^{2}
19 1+117.T+6.85e3T2 1 + 117.T + 6.85e3T^{2}
23 1+97.3T+1.21e4T2 1 + 97.3T + 1.21e4T^{2}
29 1+90.0iT−2.43e4T2 1 + 90.0iT - 2.43e4T^{2}
31 1−151.iT−2.97e4T2 1 - 151. iT - 2.97e4T^{2}
37 1+66.3T+5.06e4T2 1 + 66.3T + 5.06e4T^{2}
41 1−112.iT−6.89e4T2 1 - 112. iT - 6.89e4T^{2}
43 1−234.iT−7.95e4T2 1 - 234. iT - 7.95e4T^{2}
47 1−580.iT−1.03e5T2 1 - 580. iT - 1.03e5T^{2}
53 1+241.iT−1.48e5T2 1 + 241. iT - 1.48e5T^{2}
59 1−313.T+2.05e5T2 1 - 313.T + 2.05e5T^{2}
61 1+334.iT−2.26e5T2 1 + 334. iT - 2.26e5T^{2}
67 1+315.T+3.00e5T2 1 + 315.T + 3.00e5T^{2}
71 1−503.iT−3.57e5T2 1 - 503. iT - 3.57e5T^{2}
73 1−757.iT−3.89e5T2 1 - 757. iT - 3.89e5T^{2}
79 1−1.13e3T+4.93e5T2 1 - 1.13e3T + 4.93e5T^{2}
83 1+822.T+5.71e5T2 1 + 822.T + 5.71e5T^{2}
89 1+803.iT−7.04e5T2 1 + 803. iT - 7.04e5T^{2}
97 1−750.iT−9.12e5T2 1 - 750. 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

−11.47364142516654068641143758054, −10.34137745330191595083208506776, −9.651761812335574791902471634108, −9.069208956154922869011786998040, −8.121676673823242371189376214625, −6.33612064547317664158597688652, −5.31620062567400372905161179271, −4.21560060835508425519798966254, −2.69840881777282360107385461779, −1.81605293288851458172496694496, 0.46013426914740574840634622432, 1.89984237063254858986635722394, 3.98155114294725685901483561543, 5.34523757654177386543741998049, 6.24698298894562390161895918654, 7.10729348867012960225216834621, 7.934714935461441086867768220096, 9.021025111441561956215815490498, 10.10435286763500373279353155093, 10.50672997892905602528135333954

Graph of the ZZ-function along the critical line