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Properties

Label 2-312-104.77-c3-0-32
Degree 22
Conductor 312312
Sign −0.0966−0.995i-0.0966 - 0.995i
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  + (−1.98 + 2.01i)2-s + 3i·3-s + (−0.145 − 7.99i)4-s + 19.4·5-s + (−6.05 − 5.94i)6-s − 8.53i·7-s + (16.4 + 15.5i)8-s − 9·9-s + (−38.5 + 39.2i)10-s − 49.3·11-s + (23.9 − 0.436i)12-s + (−30.7 + 35.3i)13-s + (17.2 + 16.9i)14-s + 58.2i·15-s + (−63.9 + 2.32i)16-s + 117.·17-s + ⋯
L(s)  = 1  + (−0.700 + 0.713i)2-s + 0.577i·3-s + (−0.0181 − 0.999i)4-s + 1.73·5-s + (−0.411 − 0.404i)6-s − 0.460i·7-s + (0.726 + 0.687i)8-s − 0.333·9-s + (−1.21 + 1.24i)10-s − 1.35·11-s + (0.577 − 0.0104i)12-s + (−0.656 + 0.754i)13-s + (0.328 + 0.322i)14-s + 1.00i·15-s + (−0.999 + 0.0363i)16-s + 1.67·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.0966−0.995i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0966 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.0966−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0966 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.0966−0.995i-0.0966 - 0.995i
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), −0.0966−0.995i)(2,\ 312,\ (\ :3/2),\ -0.0966 - 0.995i)

Particular Values

L(2)L(2) ≈\approx 1.6259984291.625998429
L(12)L(\frac12) ≈\approx 1.6259984291.625998429
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+(1.98−2.01i)T 1 + (1.98 - 2.01i)T
3 1−3iT 1 - 3iT
13 1+(30.7−35.3i)T 1 + (30.7 - 35.3i)T
good5 1−19.4T+125T2 1 - 19.4T + 125T^{2}
7 1+8.53iT−343T2 1 + 8.53iT - 343T^{2}
11 1+49.3T+1.33e3T2 1 + 49.3T + 1.33e3T^{2}
17 1−117.T+4.91e3T2 1 - 117.T + 4.91e3T^{2}
19 1−26.5T+6.85e3T2 1 - 26.5T + 6.85e3T^{2}
23 1−117.T+1.21e4T2 1 - 117.T + 1.21e4T^{2}
29 1−222.iT−2.43e4T2 1 - 222. iT - 2.43e4T^{2}
31 1−240.iT−2.97e4T2 1 - 240. iT - 2.97e4T^{2}
37 1−192.T+5.06e4T2 1 - 192.T + 5.06e4T^{2}
41 1+4.97iT−6.89e4T2 1 + 4.97iT - 6.89e4T^{2}
43 1−184.iT−7.95e4T2 1 - 184. iT - 7.95e4T^{2}
47 1+410.iT−1.03e5T2 1 + 410. iT - 1.03e5T^{2}
53 1−501.iT−1.48e5T2 1 - 501. iT - 1.48e5T^{2}
59 1+353.T+2.05e5T2 1 + 353.T + 2.05e5T^{2}
61 1+268.iT−2.26e5T2 1 + 268. iT - 2.26e5T^{2}
67 1−623.T+3.00e5T2 1 - 623.T + 3.00e5T^{2}
71 1+313.iT−3.57e5T2 1 + 313. iT - 3.57e5T^{2}
73 1−254.iT−3.89e5T2 1 - 254. iT - 3.89e5T^{2}
79 1−866.T+4.93e5T2 1 - 866.T + 4.93e5T^{2}
83 1+901.T+5.71e5T2 1 + 901.T + 5.71e5T^{2}
89 1−701.iT−7.04e5T2 1 - 701. iT - 7.04e5T^{2}
97 1−531.iT−9.12e5T2 1 - 531. 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

−10.84613179085647323206478518957, −10.28220415423339957547651855093, −9.658121795261625306598626426385, −8.905502088659096942446191582316, −7.62534829571809221414858940425, −6.65970680584818620298851897616, −5.38965448427754564485884585625, −5.07968049000033226434953812066, −2.77623915075750906937957911914, −1.29511750649894900055775075103, 0.838785801215577330135936578624, 2.27440959732291163605150805336, 2.87076579484850844144756773160, 5.20697236100635569810167651402, 5.95383435502165232597141092799, 7.44509848009414898925810199506, 8.154294314223317523771442805050, 9.486701936480599324580159769734, 9.913539995138962051326913916535, 10.77039814440059133096486136599

Graph of the ZZ-function along the critical line