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Properties

Label 2-312-104.77-c3-0-34
Degree 22
Conductor 312312
Sign 0.818−0.575i0.818 - 0.575i
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  + (2.82 − 0.121i)2-s − 3i·3-s + (7.97 − 0.689i)4-s − 14.7·5-s + (−0.365 − 8.47i)6-s + 28.6i·7-s + (22.4 − 2.92i)8-s − 9·9-s + (−41.7 + 1.80i)10-s + 55.0·11-s + (−2.06 − 23.9i)12-s + (21.7 + 41.5i)13-s + (3.49 + 80.8i)14-s + 44.2i·15-s + (63.0 − 10.9i)16-s − 111.·17-s + ⋯
L(s)  = 1  + (0.999 − 0.0431i)2-s − 0.577i·3-s + (0.996 − 0.0861i)4-s − 1.32·5-s + (−0.0249 − 0.576i)6-s + 1.54i·7-s + (0.991 − 0.129i)8-s − 0.333·9-s + (−1.31 + 0.0569i)10-s + 1.50·11-s + (−0.0497 − 0.575i)12-s + (0.464 + 0.885i)13-s + (0.0666 + 1.54i)14-s + 0.762i·15-s + (0.985 − 0.171i)16-s − 1.59·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((0.818−0.575i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.575i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((0.818−0.575i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 0.818−0.575i0.818 - 0.575i
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), 0.818−0.575i)(2,\ 312,\ (\ :3/2),\ 0.818 - 0.575i)

Particular Values

L(2)L(2) ≈\approx 2.9578033952.957803395
L(12)L(\frac12) ≈\approx 2.9578033952.957803395
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+(−2.82+0.121i)T 1 + (-2.82 + 0.121i)T
3 1+3iT 1 + 3iT
13 1+(−21.7−41.5i)T 1 + (-21.7 - 41.5i)T
good5 1+14.7T+125T2 1 + 14.7T + 125T^{2}
7 1−28.6iT−343T2 1 - 28.6iT - 343T^{2}
11 1−55.0T+1.33e3T2 1 - 55.0T + 1.33e3T^{2}
17 1+111.T+4.91e3T2 1 + 111.T + 4.91e3T^{2}
19 1−80.7T+6.85e3T2 1 - 80.7T + 6.85e3T^{2}
23 1−137.T+1.21e4T2 1 - 137.T + 1.21e4T^{2}
29 1−147.iT−2.43e4T2 1 - 147. iT - 2.43e4T^{2}
31 1−231.iT−2.97e4T2 1 - 231. iT - 2.97e4T^{2}
37 1−86.8T+5.06e4T2 1 - 86.8T + 5.06e4T^{2}
41 1+122.iT−6.89e4T2 1 + 122. iT - 6.89e4T^{2}
43 1−136.iT−7.95e4T2 1 - 136. iT - 7.95e4T^{2}
47 1+554.iT−1.03e5T2 1 + 554. iT - 1.03e5T^{2}
53 1−124.iT−1.48e5T2 1 - 124. iT - 1.48e5T^{2}
59 1+348.T+2.05e5T2 1 + 348.T + 2.05e5T^{2}
61 1+674.iT−2.26e5T2 1 + 674. iT - 2.26e5T^{2}
67 1−156.T+3.00e5T2 1 - 156.T + 3.00e5T^{2}
71 1−1.14e3iT−3.57e5T2 1 - 1.14e3iT - 3.57e5T^{2}
73 1−775.iT−3.89e5T2 1 - 775. iT - 3.89e5T^{2}
79 1+1.10e3T+4.93e5T2 1 + 1.10e3T + 4.93e5T^{2}
83 1−446.T+5.71e5T2 1 - 446.T + 5.71e5T^{2}
89 1+843.iT−7.04e5T2 1 + 843. iT - 7.04e5T^{2}
97 1−321.iT−9.12e5T2 1 - 321. 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

−11.55013949967701236281307856799, −11.19433134933597363475093280746, −9.101008227696795335688279127965, −8.531471923722006204066045759329, −7.02839901553140804072349803785, −6.59596005595822683883414155148, −5.25674155549496594302466176381, −4.10993911520610966720390384016, −3.01001824948246569112926020085, −1.56560533328119244801805246904, 0.861003205208538235224899855911, 3.24724963548181546815028043988, 4.09043565697459941337668268552, 4.53944991532120110279580670462, 6.23707066789787127468021050578, 7.21451711726308701151950641840, 7.953487777313275460239050215246, 9.359497574696749723270164680620, 10.71900468418610321456709980313, 11.22557699103257324893258901608

Graph of the ZZ-function along the critical line