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Properties

Label 2-312-104.77-c3-0-10
Degree 22
Conductor 312312
Sign −0.399−0.916i-0.399 - 0.916i
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  + (−2.32 − 1.60i)2-s − 3i·3-s + (2.83 + 7.47i)4-s + 6.73·5-s + (−4.81 + 6.98i)6-s + 30.2i·7-s + (5.40 − 21.9i)8-s − 9·9-s + (−15.6 − 10.8i)10-s + 17.0·11-s + (22.4 − 8.51i)12-s + (28.4 + 37.2i)13-s + (48.6 − 70.4i)14-s − 20.1i·15-s + (−47.8 + 42.4i)16-s − 138.·17-s + ⋯
L(s)  = 1  + (−0.823 − 0.567i)2-s − 0.577i·3-s + (0.354 + 0.934i)4-s + 0.602·5-s + (−0.327 + 0.475i)6-s + 1.63i·7-s + (0.238 − 0.971i)8-s − 0.333·9-s + (−0.495 − 0.341i)10-s + 0.468·11-s + (0.539 − 0.204i)12-s + (0.606 + 0.795i)13-s + (0.928 − 1.34i)14-s − 0.347i·15-s + (−0.748 + 0.663i)16-s − 1.96·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.399−0.916i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.399−0.916i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.399−0.916i-0.399 - 0.916i
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), −0.399−0.916i)(2,\ 312,\ (\ :3/2),\ -0.399 - 0.916i)

Particular Values

L(2)L(2) ≈\approx 0.47014159100.4701415910
L(12)L(\frac12) ≈\approx 0.47014159100.4701415910
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+(2.32+1.60i)T 1 + (2.32 + 1.60i)T
3 1+3iT 1 + 3iT
13 1+(−28.4−37.2i)T 1 + (-28.4 - 37.2i)T
good5 1−6.73T+125T2 1 - 6.73T + 125T^{2}
7 1−30.2iT−343T2 1 - 30.2iT - 343T^{2}
11 1−17.0T+1.33e3T2 1 - 17.0T + 1.33e3T^{2}
17 1+138.T+4.91e3T2 1 + 138.T + 4.91e3T^{2}
19 1+72.5T+6.85e3T2 1 + 72.5T + 6.85e3T^{2}
23 1+90.4T+1.21e4T2 1 + 90.4T + 1.21e4T^{2}
29 1+265.iT−2.43e4T2 1 + 265. iT - 2.43e4T^{2}
31 1+37.6iT−2.97e4T2 1 + 37.6iT - 2.97e4T^{2}
37 1+274.T+5.06e4T2 1 + 274.T + 5.06e4T^{2}
41 1−70.0iT−6.89e4T2 1 - 70.0iT - 6.89e4T^{2}
43 1+273.iT−7.95e4T2 1 + 273. iT - 7.95e4T^{2}
47 1−419.iT−1.03e5T2 1 - 419. iT - 1.03e5T^{2}
53 1−426.iT−1.48e5T2 1 - 426. iT - 1.48e5T^{2}
59 1+162.T+2.05e5T2 1 + 162.T + 2.05e5T^{2}
61 1−629.iT−2.26e5T2 1 - 629. iT - 2.26e5T^{2}
67 1−180.T+3.00e5T2 1 - 180.T + 3.00e5T^{2}
71 1+690.iT−3.57e5T2 1 + 690. iT - 3.57e5T^{2}
73 1+63.7iT−3.89e5T2 1 + 63.7iT - 3.89e5T^{2}
79 1−27.2T+4.93e5T2 1 - 27.2T + 4.93e5T^{2}
83 1−1.09e3T+5.71e5T2 1 - 1.09e3T + 5.71e5T^{2}
89 1−1.26e3iT−7.04e5T2 1 - 1.26e3iT - 7.04e5T^{2}
97 1−580.iT−9.12e5T2 1 - 580. 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

−11.66546009052174832122447032299, −10.68338861840674003490675657339, −9.297347039596442938961608800981, −8.966015042821321267387107723453, −8.070748946384714455117533109128, −6.55495783560418444990377119623, −6.05971802256525659347150118444, −4.18530655723606770343676873690, −2.35671334810114944794946861767, −1.91056232001491363005832838818, 0.20877153900720032172683168041, 1.78273431053852773053469524239, 3.80069290010843514383506780625, 4.96831484530880782434563315883, 6.31678301007848125818037689269, 6.97307909955135056042237750966, 8.248635084910105236312057342725, 9.034915681055426192359350825605, 10.12488737669841436052839503809, 10.60728232013908109546068886904

Graph of the ZZ-function along the critical line