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Properties

Label 2-312-104.77-c3-0-33
Degree 22
Conductor 312312
Sign 0.859−0.510i0.859 - 0.510i
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  + (−2.09 − 1.90i)2-s + 3i·3-s + (0.748 + 7.96i)4-s + 5.98·5-s + (5.71 − 6.27i)6-s + 13.5i·7-s + (13.5 − 18.0i)8-s − 9·9-s + (−12.5 − 11.3i)10-s + 38.8·11-s + (−23.8 + 2.24i)12-s + (17.8 − 43.3i)13-s + (25.8 − 28.3i)14-s + 17.9i·15-s + (−62.8 + 11.9i)16-s + 10.7·17-s + ⋯
L(s)  = 1  + (−0.739 − 0.673i)2-s + 0.577i·3-s + (0.0935 + 0.995i)4-s + 0.535·5-s + (0.388 − 0.426i)6-s + 0.732i·7-s + (0.601 − 0.799i)8-s − 0.333·9-s + (−0.395 − 0.360i)10-s + 1.06·11-s + (−0.574 + 0.0540i)12-s + (0.380 − 0.924i)13-s + (0.493 − 0.541i)14-s + 0.308i·15-s + (−0.982 + 0.186i)16-s + 0.153·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.859−0.510i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.859−0.510i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.859−0.510i0.859 - 0.510i
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), 0.859−0.510i)(2,\ 312,\ (\ :3/2),\ 0.859 - 0.510i)

Particular Values

L(2)L(2) ≈\approx 1.4694896441.469489644
L(12)L(\frac12) ≈\approx 1.4694896441.469489644
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+(2.09+1.90i)T 1 + (2.09 + 1.90i)T
3 1−3iT 1 - 3iT
13 1+(−17.8+43.3i)T 1 + (-17.8 + 43.3i)T
good5 1−5.98T+125T2 1 - 5.98T + 125T^{2}
7 1−13.5iT−343T2 1 - 13.5iT - 343T^{2}
11 1−38.8T+1.33e3T2 1 - 38.8T + 1.33e3T^{2}
17 1−10.7T+4.91e3T2 1 - 10.7T + 4.91e3T^{2}
19 1−50.0T+6.85e3T2 1 - 50.0T + 6.85e3T^{2}
23 1−58.0T+1.21e4T2 1 - 58.0T + 1.21e4T^{2}
29 1−140.iT−2.43e4T2 1 - 140. iT - 2.43e4T^{2}
31 1−49.2iT−2.97e4T2 1 - 49.2iT - 2.97e4T^{2}
37 1−109.T+5.06e4T2 1 - 109.T + 5.06e4T^{2}
41 1−200.iT−6.89e4T2 1 - 200. iT - 6.89e4T^{2}
43 1−53.5iT−7.95e4T2 1 - 53.5iT - 7.95e4T^{2}
47 1−95.3iT−1.03e5T2 1 - 95.3iT - 1.03e5T^{2}
53 1−385.iT−1.48e5T2 1 - 385. iT - 1.48e5T^{2}
59 1−305.T+2.05e5T2 1 - 305.T + 2.05e5T^{2}
61 1−164.iT−2.26e5T2 1 - 164. iT - 2.26e5T^{2}
67 1−962.T+3.00e5T2 1 - 962.T + 3.00e5T^{2}
71 1+195.iT−3.57e5T2 1 + 195. iT - 3.57e5T^{2}
73 1+317.iT−3.89e5T2 1 + 317. iT - 3.89e5T^{2}
79 1+221.T+4.93e5T2 1 + 221.T + 4.93e5T^{2}
83 1−445.T+5.71e5T2 1 - 445.T + 5.71e5T^{2}
89 1−523.iT−7.04e5T2 1 - 523. iT - 7.04e5T^{2}
97 1−147.iT−9.12e5T2 1 - 147. 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

−11.21134155262245655947271730088, −10.29240086876063117644557363074, −9.431831510962932626652739838940, −8.879474285033143345829764150546, −7.82050673187444742489914277715, −6.44055748069644478768753340046, −5.27591898346481411366751168289, −3.78756435599315693963559721972, −2.71747755373699698729287805497, −1.20755407090072979891103423082, 0.839364575551442153096040475399, 1.95549895716410814754300699083, 4.05680582004683615166751939017, 5.56886203315045972951755443732, 6.55160106545179815221942918544, 7.17342835573860584688802977576, 8.257395421369844316372558973403, 9.277368849927655830647621461775, 9.916512162522840508832480469761, 11.13120974560011646303339402236

Graph of the ZZ-function along the critical line