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Properties

Label 2-312-104.77-c3-0-36
Degree 22
Conductor 312312
Sign 0.580+0.814i0.580 + 0.814i
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  + (−1.62 + 2.31i)2-s − 3i·3-s + (−2.71 − 7.52i)4-s − 20.0·5-s + (6.94 + 4.87i)6-s + 33.1i·7-s + (21.8 + 5.93i)8-s − 9·9-s + (32.5 − 46.4i)10-s + 7.50·11-s + (−22.5 + 8.15i)12-s + (−29.7 + 36.2i)13-s + (−76.7 − 53.8i)14-s + 60.1i·15-s + (−49.2 + 40.9i)16-s + 88.2·17-s + ⋯
L(s)  = 1  + (−0.574 + 0.818i)2-s − 0.577i·3-s + (−0.339 − 0.940i)4-s − 1.79·5-s + (0.472 + 0.331i)6-s + 1.78i·7-s + (0.964 + 0.262i)8-s − 0.333·9-s + (1.03 − 1.46i)10-s + 0.205·11-s + (−0.543 + 0.196i)12-s + (−0.633 + 0.773i)13-s + (−1.46 − 1.02i)14-s + 1.03i·15-s + (−0.769 + 0.639i)16-s + 1.25·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.580+0.814i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.580+0.814i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.580+0.814i0.580 + 0.814i
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), 0.580+0.814i)(2,\ 312,\ (\ :3/2),\ 0.580 + 0.814i)

Particular Values

L(2)L(2) ≈\approx 0.30616818290.3061681829
L(12)L(\frac12) ≈\approx 0.30616818290.3061681829
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+(1.62−2.31i)T 1 + (1.62 - 2.31i)T
3 1+3iT 1 + 3iT
13 1+(29.7−36.2i)T 1 + (29.7 - 36.2i)T
good5 1+20.0T+125T2 1 + 20.0T + 125T^{2}
7 1−33.1iT−343T2 1 - 33.1iT - 343T^{2}
11 1−7.50T+1.33e3T2 1 - 7.50T + 1.33e3T^{2}
17 1−88.2T+4.91e3T2 1 - 88.2T + 4.91e3T^{2}
19 1+43.0T+6.85e3T2 1 + 43.0T + 6.85e3T^{2}
23 1+192.T+1.21e4T2 1 + 192.T + 1.21e4T^{2}
29 1+133.iT−2.43e4T2 1 + 133. iT - 2.43e4T^{2}
31 1+114.iT−2.97e4T2 1 + 114. iT - 2.97e4T^{2}
37 1+59.6T+5.06e4T2 1 + 59.6T + 5.06e4T^{2}
41 1+505.iT−6.89e4T2 1 + 505. iT - 6.89e4T^{2}
43 1−142.iT−7.95e4T2 1 - 142. iT - 7.95e4T^{2}
47 1−97.2iT−1.03e5T2 1 - 97.2iT - 1.03e5T^{2}
53 1+141.iT−1.48e5T2 1 + 141. iT - 1.48e5T^{2}
59 1−794.T+2.05e5T2 1 - 794.T + 2.05e5T^{2}
61 1+124.iT−2.26e5T2 1 + 124. iT - 2.26e5T^{2}
67 1−174.T+3.00e5T2 1 - 174.T + 3.00e5T^{2}
71 1+519.iT−3.57e5T2 1 + 519. iT - 3.57e5T^{2}
73 1−616.iT−3.89e5T2 1 - 616. iT - 3.89e5T^{2}
79 1−937.T+4.93e5T2 1 - 937.T + 4.93e5T^{2}
83 1+1.14e3T+5.71e5T2 1 + 1.14e3T + 5.71e5T^{2}
89 1+15.8iT−7.04e5T2 1 + 15.8iT - 7.04e5T^{2}
97 1−481.iT−9.12e5T2 1 - 481. 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

−11.41955932198992060415819452813, −9.885852090845465862997375413131, −8.810091560858840855663683832173, −8.145085704381023186366153719456, −7.48640637171887984391151094749, −6.35631574649420557245787892110, −5.35635971193052173852542274783, −4.00869392754295006160156270524, −2.17797659682752133883260165592, −0.19377890161410727837721888264, 0.871673197118746359271607380181, 3.31306934738544358187255432533, 3.88449788971339079094683441742, 4.73063234444842749042329282155, 7.08157595550907626811304158607, 7.80569403461171258238389131636, 8.376483940315759307022547620621, 9.964253457650350038394627941844, 10.39206071674242216222629301601, 11.25031747195643483468856702493

Graph of the ZZ-function along the critical line