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Properties

Label 2-312-104.77-c3-0-22
Degree 22
Conductor 312312
Sign 0.250−0.968i0.250 - 0.968i
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  + (−2.81 − 0.274i)2-s + 3i·3-s + (7.84 + 1.54i)4-s − 8.71·5-s + (0.824 − 8.44i)6-s − 16.6i·7-s + (−21.6 − 6.51i)8-s − 9·9-s + (24.5 + 2.39i)10-s − 35.9·11-s + (−4.64 + 23.5i)12-s + (46.8 − 1.83i)13-s + (−4.58 + 46.9i)14-s − 26.1i·15-s + (59.2 + 24.2i)16-s + 23.1·17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0971i)2-s + 0.577i·3-s + (0.981 + 0.193i)4-s − 0.779·5-s + (0.0560 − 0.574i)6-s − 0.899i·7-s + (−0.957 − 0.287i)8-s − 0.333·9-s + (0.775 + 0.0757i)10-s − 0.985·11-s + (−0.111 + 0.566i)12-s + (0.999 − 0.0392i)13-s + (−0.0874 + 0.895i)14-s − 0.450i·15-s + (0.925 + 0.379i)16-s + 0.330·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.250−0.968i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 - 0.968i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.250−0.968i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.250−0.968i0.250 - 0.968i
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), 0.250−0.968i)(2,\ 312,\ (\ :3/2),\ 0.250 - 0.968i)

Particular Values

L(2)L(2) ≈\approx 0.72339174650.7233917465
L(12)L(\frac12) ≈\approx 0.72339174650.7233917465
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+(2.81+0.274i)T 1 + (2.81 + 0.274i)T
3 1−3iT 1 - 3iT
13 1+(−46.8+1.83i)T 1 + (-46.8 + 1.83i)T
good5 1+8.71T+125T2 1 + 8.71T + 125T^{2}
7 1+16.6iT−343T2 1 + 16.6iT - 343T^{2}
11 1+35.9T+1.33e3T2 1 + 35.9T + 1.33e3T^{2}
17 1−23.1T+4.91e3T2 1 - 23.1T + 4.91e3T^{2}
19 1−30.2T+6.85e3T2 1 - 30.2T + 6.85e3T^{2}
23 1−72.9T+1.21e4T2 1 - 72.9T + 1.21e4T^{2}
29 1−119.iT−2.43e4T2 1 - 119. iT - 2.43e4T^{2}
31 1−96.3iT−2.97e4T2 1 - 96.3iT - 2.97e4T^{2}
37 1+33.6T+5.06e4T2 1 + 33.6T + 5.06e4T^{2}
41 1+60.2iT−6.89e4T2 1 + 60.2iT - 6.89e4T^{2}
43 1+6.93iT−7.95e4T2 1 + 6.93iT - 7.95e4T^{2}
47 1−452.iT−1.03e5T2 1 - 452. iT - 1.03e5T^{2}
53 1−308.iT−1.48e5T2 1 - 308. iT - 1.48e5T^{2}
59 1−92.9T+2.05e5T2 1 - 92.9T + 2.05e5T^{2}
61 1−538.iT−2.26e5T2 1 - 538. iT - 2.26e5T^{2}
67 1−484.T+3.00e5T2 1 - 484.T + 3.00e5T^{2}
71 1−447.iT−3.57e5T2 1 - 447. iT - 3.57e5T^{2}
73 1−763.iT−3.89e5T2 1 - 763. iT - 3.89e5T^{2}
79 1−354.T+4.93e5T2 1 - 354.T + 4.93e5T^{2}
83 1+232.T+5.71e5T2 1 + 232.T + 5.71e5T^{2}
89 1−1.31e3iT−7.04e5T2 1 - 1.31e3iT - 7.04e5T^{2}
97 1−1.33e3iT−9.12e5T2 1 - 1.33e3iT - 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

−10.95673467676045585229285284442, −10.66719278581220763827302905339, −9.636860969251879299106716987755, −8.562922349116070354793611043013, −7.79378910709490481440840609221, −6.95264114034795948551947459807, −5.53211853719303259558683578484, −4.01297662320083833710811422273, −3.01155174402872136269377904080, −0.998805556098292689700177974441, 0.46285098012106068700911004779, 2.08742294182894605063830047604, 3.34045602431170568087551565846, 5.37311149789673425737128064756, 6.32312730891420545831423894117, 7.51306319717005783362337957816, 8.157867656215473669372942217397, 8.914033128086512888030796725492, 10.05274612496076816654442212425, 11.18122013690879023038178561089

Graph of the ZZ-function along the critical line