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Properties

Label 2-312-104.77-c3-0-23
Degree 22
Conductor 312312
Sign −0.991−0.131i-0.991 - 0.131i
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  + (−2.30 + 1.63i)2-s + 3i·3-s + (2.62 − 7.55i)4-s − 3.15·5-s + (−4.91 − 6.91i)6-s + 31.8i·7-s + (6.33 + 21.7i)8-s − 9·9-s + (7.28 − 5.17i)10-s + 30.3·11-s + (22.6 + 7.87i)12-s + (42.8 + 18.9i)13-s + (−52.1 − 73.3i)14-s − 9.47i·15-s + (−50.2 − 39.6i)16-s + 33.4·17-s + ⋯
L(s)  = 1  + (−0.814 + 0.579i)2-s + 0.577i·3-s + (0.328 − 0.944i)4-s − 0.282·5-s + (−0.334 − 0.470i)6-s + 1.71i·7-s + (0.280 + 0.959i)8-s − 0.333·9-s + (0.230 − 0.163i)10-s + 0.831·11-s + (0.545 + 0.189i)12-s + (0.914 + 0.403i)13-s + (−0.995 − 1.40i)14-s − 0.163i·15-s + (−0.784 − 0.620i)16-s + 0.477·17-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=((−0.991−0.131i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=((−0.991−0.131i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: −0.991−0.131i-0.991 - 0.131i
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), −0.991−0.131i)(2,\ 312,\ (\ :3/2),\ -0.991 - 0.131i)

Particular Values

L(2)L(2) ≈\approx 1.0275683421.027568342
L(12)L(\frac12) ≈\approx 1.0275683421.027568342
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+(2.30−1.63i)T 1 + (2.30 - 1.63i)T
3 1−3iT 1 - 3iT
13 1+(−42.8−18.9i)T 1 + (-42.8 - 18.9i)T
good5 1+3.15T+125T2 1 + 3.15T + 125T^{2}
7 1−31.8iT−343T2 1 - 31.8iT - 343T^{2}
11 1−30.3T+1.33e3T2 1 - 30.3T + 1.33e3T^{2}
17 1−33.4T+4.91e3T2 1 - 33.4T + 4.91e3T^{2}
19 1−32.4T+6.85e3T2 1 - 32.4T + 6.85e3T^{2}
23 1−92.1T+1.21e4T2 1 - 92.1T + 1.21e4T^{2}
29 1−11.6iT−2.43e4T2 1 - 11.6iT - 2.43e4T^{2}
31 1−328.iT−2.97e4T2 1 - 328. iT - 2.97e4T^{2}
37 1+271.T+5.06e4T2 1 + 271.T + 5.06e4T^{2}
41 1−153.iT−6.89e4T2 1 - 153. iT - 6.89e4T^{2}
43 1−26.1iT−7.95e4T2 1 - 26.1iT - 7.95e4T^{2}
47 1+72.2iT−1.03e5T2 1 + 72.2iT - 1.03e5T^{2}
53 1+666.iT−1.48e5T2 1 + 666. iT - 1.48e5T^{2}
59 1+512.T+2.05e5T2 1 + 512.T + 2.05e5T^{2}
61 1−527.iT−2.26e5T2 1 - 527. iT - 2.26e5T^{2}
67 1+863.T+3.00e5T2 1 + 863.T + 3.00e5T^{2}
71 1+810.iT−3.57e5T2 1 + 810. iT - 3.57e5T^{2}
73 1−157.iT−3.89e5T2 1 - 157. iT - 3.89e5T^{2}
79 1−796.T+4.93e5T2 1 - 796.T + 4.93e5T^{2}
83 1+69.5T+5.71e5T2 1 + 69.5T + 5.71e5T^{2}
89 1−1.63e3iT−7.04e5T2 1 - 1.63e3iT - 7.04e5T^{2}
97 1+904.iT−9.12e5T2 1 + 904. 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

−11.60991797118290477922401603321, −10.63383622784798505744943152787, −9.450014833855010997676675180181, −8.916023280722359084307607312523, −8.233358921504624572575301658212, −6.79634486975057073773830967905, −5.86305038718010086359414404583, −4.98247568220220865464294000133, −3.26206622350318468171778908791, −1.60744598075888426063148549028, 0.54493420098075271408603402426, 1.46332341580153686613683588306, 3.34950553880089636398630112727, 4.15868131237184831652942986795, 6.21748198319445463690786530278, 7.33006061554603667371430016224, 7.76704615303356423229135589488, 8.934578491759418462475888892257, 9.963615990521478754772273295794, 10.86235408483314418198233437155

Graph of the ZZ-function along the critical line