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Properties

Label 2-312-104.77-c3-0-26
Degree 22
Conductor 312312
Sign 0.292−0.956i0.292 - 0.956i
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  + (0.0631 + 2.82i)2-s − 3i·3-s + (−7.99 + 0.357i)4-s − 11.9·5-s + (8.48 − 0.189i)6-s − 11.1i·7-s + (−1.51 − 22.5i)8-s − 9·9-s + (−0.756 − 33.8i)10-s − 24.4·11-s + (1.07 + 23.9i)12-s + (−16.6 + 43.8i)13-s + (31.6 − 0.706i)14-s + 35.9i·15-s + (63.7 − 5.70i)16-s + 68.2·17-s + ⋯
L(s)  = 1  + (0.0223 + 0.999i)2-s − 0.577i·3-s + (−0.999 + 0.0446i)4-s − 1.07·5-s + (0.577 − 0.0128i)6-s − 0.604i·7-s + (−0.0669 − 0.997i)8-s − 0.333·9-s + (−0.0239 − 1.07i)10-s − 0.669·11-s + (0.0257 + 0.576i)12-s + (−0.355 + 0.934i)13-s + (0.604 − 0.0134i)14-s + 0.618i·15-s + (0.996 − 0.0891i)16-s + 0.974·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.292−0.956i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.292−0.956i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.292−0.956i0.292 - 0.956i
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), 0.292−0.956i)(2,\ 312,\ (\ :3/2),\ 0.292 - 0.956i)

Particular Values

L(2)L(2) ≈\approx 1.0562672201.056267220
L(12)L(\frac12) ≈\approx 1.0562672201.056267220
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+(−0.0631−2.82i)T 1 + (-0.0631 - 2.82i)T
3 1+3iT 1 + 3iT
13 1+(16.6−43.8i)T 1 + (16.6 - 43.8i)T
good5 1+11.9T+125T2 1 + 11.9T + 125T^{2}
7 1+11.1iT−343T2 1 + 11.1iT - 343T^{2}
11 1+24.4T+1.33e3T2 1 + 24.4T + 1.33e3T^{2}
17 1−68.2T+4.91e3T2 1 - 68.2T + 4.91e3T^{2}
19 1−29.3T+6.85e3T2 1 - 29.3T + 6.85e3T^{2}
23 1−197.T+1.21e4T2 1 - 197.T + 1.21e4T^{2}
29 1−184.iT−2.43e4T2 1 - 184. iT - 2.43e4T^{2}
31 1+32.3iT−2.97e4T2 1 + 32.3iT - 2.97e4T^{2}
37 1−81.8T+5.06e4T2 1 - 81.8T + 5.06e4T^{2}
41 1−159.iT−6.89e4T2 1 - 159. iT - 6.89e4T^{2}
43 1+400.iT−7.95e4T2 1 + 400. iT - 7.95e4T^{2}
47 1−183.iT−1.03e5T2 1 - 183. iT - 1.03e5T^{2}
53 1−220.iT−1.48e5T2 1 - 220. iT - 1.48e5T^{2}
59 1−464.T+2.05e5T2 1 - 464.T + 2.05e5T^{2}
61 1−182.iT−2.26e5T2 1 - 182. iT - 2.26e5T^{2}
67 1+292.T+3.00e5T2 1 + 292.T + 3.00e5T^{2}
71 1−913.iT−3.57e5T2 1 - 913. iT - 3.57e5T^{2}
73 1+300.iT−3.89e5T2 1 + 300. iT - 3.89e5T^{2}
79 1+61.6T+4.93e5T2 1 + 61.6T + 4.93e5T^{2}
83 1−381.T+5.71e5T2 1 - 381.T + 5.71e5T^{2}
89 1+172.iT−7.04e5T2 1 + 172. iT - 7.04e5T^{2}
97 1+933.iT−9.12e5T2 1 + 933. 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

−11.59360644686961799736764004594, −10.47044247302731738024323367772, −9.238760211121158247772420434840, −8.269532713149982705653952532168, −7.31751729998919280187321467982, −7.05163277700182162672599948190, −5.53052006007092475726718159284, −4.47260867472108062548856580739, −3.28236511013811096299323105289, −0.873581271658875457501815729818, 0.57501740409025770203239732198, 2.71245713261597840956784609074, 3.53906435177520969497534979974, 4.80647776535702995879217479533, 5.58973340132067312749575438354, 7.62439704519346621585300980417, 8.321951507165944838208766995279, 9.392934503917987432752847616051, 10.25138423640471604559609172088, 11.11525704040506633920349460638

Graph of the ZZ-function along the critical line