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Properties

Label 2-312-104.77-c3-0-17
Degree 22
Conductor 312312
Sign 0.839−0.542i0.839 - 0.542i
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  + (−1.61 − 2.32i)2-s − 3i·3-s + (−2.78 + 7.50i)4-s + 1.96·5-s + (−6.96 + 4.84i)6-s − 7.12i·7-s + (21.9 − 5.66i)8-s − 9·9-s + (−3.16 − 4.55i)10-s − 39.6·11-s + (22.5 + 8.34i)12-s + (14.7 + 44.4i)13-s + (−16.5 + 11.5i)14-s − 5.88i·15-s + (−48.5 − 41.7i)16-s − 20.3·17-s + ⋯
L(s)  = 1  + (−0.571 − 0.820i)2-s − 0.577i·3-s + (−0.347 + 0.937i)4-s + 0.175·5-s + (−0.473 + 0.329i)6-s − 0.384i·7-s + (0.968 − 0.250i)8-s − 0.333·9-s + (−0.100 − 0.144i)10-s − 1.08·11-s + (0.541 + 0.200i)12-s + (0.315 + 0.948i)13-s + (−0.315 + 0.219i)14-s − 0.101i·15-s + (−0.758 − 0.651i)16-s − 0.291·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.839−0.542i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.839−0.542i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.839−0.542i0.839 - 0.542i
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), 0.839−0.542i)(2,\ 312,\ (\ :3/2),\ 0.839 - 0.542i)

Particular Values

L(2)L(2) ≈\approx 0.71331789460.7133178946
L(12)L(\frac12) ≈\approx 0.71331789460.7133178946
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+(1.61+2.32i)T 1 + (1.61 + 2.32i)T
3 1+3iT 1 + 3iT
13 1+(−14.7−44.4i)T 1 + (-14.7 - 44.4i)T
good5 1−1.96T+125T2 1 - 1.96T + 125T^{2}
7 1+7.12iT−343T2 1 + 7.12iT - 343T^{2}
11 1+39.6T+1.33e3T2 1 + 39.6T + 1.33e3T^{2}
17 1+20.3T+4.91e3T2 1 + 20.3T + 4.91e3T^{2}
19 1−78.7T+6.85e3T2 1 - 78.7T + 6.85e3T^{2}
23 1+108.T+1.21e4T2 1 + 108.T + 1.21e4T^{2}
29 1−306.iT−2.43e4T2 1 - 306. iT - 2.43e4T^{2}
31 1−122.iT−2.97e4T2 1 - 122. iT - 2.97e4T^{2}
37 1−238.T+5.06e4T2 1 - 238.T + 5.06e4T^{2}
41 1−113.iT−6.89e4T2 1 - 113. iT - 6.89e4T^{2}
43 1−443.iT−7.95e4T2 1 - 443. iT - 7.95e4T^{2}
47 1+435.iT−1.03e5T2 1 + 435. iT - 1.03e5T^{2}
53 1+496.iT−1.48e5T2 1 + 496. iT - 1.48e5T^{2}
59 1−868.T+2.05e5T2 1 - 868.T + 2.05e5T^{2}
61 1−355.iT−2.26e5T2 1 - 355. iT - 2.26e5T^{2}
67 1+792.T+3.00e5T2 1 + 792.T + 3.00e5T^{2}
71 1+208.iT−3.57e5T2 1 + 208. iT - 3.57e5T^{2}
73 1−1.07e3iT−3.89e5T2 1 - 1.07e3iT - 3.89e5T^{2}
79 1−1.22e3T+4.93e5T2 1 - 1.22e3T + 4.93e5T^{2}
83 1+701.T+5.71e5T2 1 + 701.T + 5.71e5T^{2}
89 1−14.2iT−7.04e5T2 1 - 14.2iT - 7.04e5T^{2}
97 1−1.16e3iT−9.12e5T2 1 - 1.16e3iT - 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

−11.35065486631765691353245851847, −10.42084915929280399317000585529, −9.573439856570184047806313444596, −8.519536800097635106584575841743, −7.66796210710607373256317197129, −6.75670391082269502393012231951, −5.22171390866517861031667428914, −3.81935324596234666655728866274, −2.49965439163537564670914243944, −1.28581123225543430000125773896, 0.33590818090604311821022457188, 2.41965390304137052393937754127, 4.19597074195738092520929621361, 5.54360285678784686736392194274, 5.94995907545602359554610734230, 7.63464408454512468432426796427, 8.140416280802092304621334641135, 9.348613688996158780676919053423, 10.02212181201847391274902334783, 10.79941850948790617531010424126

Graph of the ZZ-function along the critical line