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Properties

Label 2-312-104.77-c3-0-3
Degree 22
Conductor 312312
Sign −0.992−0.119i-0.992 - 0.119i
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  + (2.80 − 0.390i)2-s + 3i·3-s + (7.69 − 2.18i)4-s − 14.7·5-s + (1.17 + 8.40i)6-s − 0.217i·7-s + (20.6 − 9.14i)8-s − 9·9-s + (−41.4 + 5.78i)10-s − 43.4·11-s + (6.56 + 23.0i)12-s + (−23.9 + 40.2i)13-s + (−0.0851 − 0.609i)14-s − 44.3i·15-s + (54.4 − 33.6i)16-s − 16.1·17-s + ⋯
L(s)  = 1  + (0.990 − 0.138i)2-s + 0.577i·3-s + (0.961 − 0.273i)4-s − 1.32·5-s + (0.0797 + 0.571i)6-s − 0.0117i·7-s + (0.914 − 0.403i)8-s − 0.333·9-s + (−1.31 + 0.182i)10-s − 1.19·11-s + (0.158 + 0.555i)12-s + (−0.510 + 0.859i)13-s + (−0.00162 − 0.0116i)14-s − 0.763i·15-s + (0.850 − 0.526i)16-s − 0.230·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.992−0.119i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.992−0.119i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.992−0.119i-0.992 - 0.119i
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), −0.992−0.119i)(2,\ 312,\ (\ :3/2),\ -0.992 - 0.119i)

Particular Values

L(2)L(2) ≈\approx 0.48659188740.4865918874
L(12)L(\frac12) ≈\approx 0.48659188740.4865918874
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+(−2.80+0.390i)T 1 + (-2.80 + 0.390i)T
3 1−3iT 1 - 3iT
13 1+(23.9−40.2i)T 1 + (23.9 - 40.2i)T
good5 1+14.7T+125T2 1 + 14.7T + 125T^{2}
7 1+0.217iT−343T2 1 + 0.217iT - 343T^{2}
11 1+43.4T+1.33e3T2 1 + 43.4T + 1.33e3T^{2}
17 1+16.1T+4.91e3T2 1 + 16.1T + 4.91e3T^{2}
19 1+72.7T+6.85e3T2 1 + 72.7T + 6.85e3T^{2}
23 1+83.4T+1.21e4T2 1 + 83.4T + 1.21e4T^{2}
29 1+128.iT−2.43e4T2 1 + 128. iT - 2.43e4T^{2}
31 1−306.iT−2.97e4T2 1 - 306. iT - 2.97e4T^{2}
37 1+340.T+5.06e4T2 1 + 340.T + 5.06e4T^{2}
41 1+401.iT−6.89e4T2 1 + 401. iT - 6.89e4T^{2}
43 1−32.3iT−7.95e4T2 1 - 32.3iT - 7.95e4T^{2}
47 1+118.iT−1.03e5T2 1 + 118. iT - 1.03e5T^{2}
53 1−486.iT−1.48e5T2 1 - 486. iT - 1.48e5T^{2}
59 1−241.T+2.05e5T2 1 - 241.T + 2.05e5T^{2}
61 1−457.iT−2.26e5T2 1 - 457. iT - 2.26e5T^{2}
67 1−121.T+3.00e5T2 1 - 121.T + 3.00e5T^{2}
71 1+251.iT−3.57e5T2 1 + 251. iT - 3.57e5T^{2}
73 1+66.4iT−3.89e5T2 1 + 66.4iT - 3.89e5T^{2}
79 1+164.T+4.93e5T2 1 + 164.T + 4.93e5T^{2}
83 1−311.T+5.71e5T2 1 - 311.T + 5.71e5T^{2}
89 1−1.00e3iT−7.04e5T2 1 - 1.00e3iT - 7.04e5T^{2}
97 1+433.iT−9.12e5T2 1 + 433. 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

−11.95122695062350058993543790867, −10.79936548952409557581244534656, −10.32335131525393440817835223081, −8.756358074101324365609216070236, −7.70813541432929599972885148537, −6.81602403683991170762683382235, −5.39929231951182122851024569595, −4.44554934733360292466231589157, −3.68614109256587282194325633010, −2.35836947927465205586263934305, 0.11557005335099657550168895234, 2.32518423693991638623638742992, 3.48676385687586487428467353269, 4.65274039811674551831408512002, 5.71141083666505819710122428781, 6.94629998305340678279329914167, 7.82217777308566188050932878190, 8.269869128131003227402052287144, 10.24865484095014834703059034885, 11.13308269811215087092146284664

Graph of the ZZ-function along the critical line