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Properties

Label 2-810-5.4-c3-0-30
Degree 22
Conductor 810810
Sign −0.953+0.301i-0.953 + 0.301i
Analytic cond. 47.791547.7915
Root an. cond. 6.913146.91314
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  + 2i·2-s − 4·4-s + (3.36 + 10.6i)5-s + 27.8i·7-s − 8i·8-s + (−21.3 + 6.73i)10-s + 13.2·11-s + 54.4i·13-s − 55.6·14-s + 16·16-s + 28.7i·17-s + 109.·19-s + (−13.4 − 42.6i)20-s + 26.4i·22-s + 176. i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.301 + 0.953i)5-s + 1.50i·7-s − 0.353i·8-s + (−0.674 + 0.212i)10-s + 0.361·11-s + 1.16i·13-s − 1.06·14-s + 0.250·16-s + 0.410i·17-s + 1.32·19-s + (−0.150 − 0.476i)20-s + 0.255i·22-s + 1.60i·23-s + ⋯

Functional equation

Λ(s)=(810s/2 ΓC(s) L(s)=((−0.953+0.301i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(810s/2 ΓC(s+3/2) L(s)=((−0.953+0.301i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 810810    =    2⋅34⋅52 \cdot 3^{4} \cdot 5
Sign: −0.953+0.301i-0.953 + 0.301i
Analytic conductor: 47.791547.7915
Root analytic conductor: 6.913146.91314
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ810(649,⋅)\chi_{810} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 810, ( :3/2), −0.953+0.301i)(2,\ 810,\ (\ :3/2),\ -0.953 + 0.301i)

Particular Values

L(2)L(2) ≈\approx 1.9789345841.978934584
L(12)L(\frac12) ≈\approx 1.9789345841.978934584
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 1−2iT 1 - 2iT
3 1 1
5 1+(−3.36−10.6i)T 1 + (-3.36 - 10.6i)T
good7 1−27.8iT−343T2 1 - 27.8iT - 343T^{2}
11 1−13.2T+1.33e3T2 1 - 13.2T + 1.33e3T^{2}
13 1−54.4iT−2.19e3T2 1 - 54.4iT - 2.19e3T^{2}
17 1−28.7iT−4.91e3T2 1 - 28.7iT - 4.91e3T^{2}
19 1−109.T+6.85e3T2 1 - 109.T + 6.85e3T^{2}
23 1−176.iT−1.21e4T2 1 - 176. iT - 1.21e4T^{2}
29 1−68.8T+2.43e4T2 1 - 68.8T + 2.43e4T^{2}
31 1+11.3T+2.97e4T2 1 + 11.3T + 2.97e4T^{2}
37 1+73.9iT−5.06e4T2 1 + 73.9iT - 5.06e4T^{2}
41 1−320.T+6.89e4T2 1 - 320.T + 6.89e4T^{2}
43 1+403.iT−7.95e4T2 1 + 403. iT - 7.95e4T^{2}
47 1−400.iT−1.03e5T2 1 - 400. iT - 1.03e5T^{2}
53 1+106.iT−1.48e5T2 1 + 106. iT - 1.48e5T^{2}
59 1−434.T+2.05e5T2 1 - 434.T + 2.05e5T^{2}
61 1−812.T+2.26e5T2 1 - 812.T + 2.26e5T^{2}
67 1−411.iT−3.00e5T2 1 - 411. iT - 3.00e5T^{2}
71 1+254.T+3.57e5T2 1 + 254.T + 3.57e5T^{2}
73 1+586.iT−3.89e5T2 1 + 586. iT - 3.89e5T^{2}
79 1+1.35e3T+4.93e5T2 1 + 1.35e3T + 4.93e5T^{2}
83 1+752.iT−5.71e5T2 1 + 752. iT - 5.71e5T^{2}
89 1+1.55e3T+7.04e5T2 1 + 1.55e3T + 7.04e5T^{2}
97 1+816.iT−9.12e5T2 1 + 816. 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

−10.01039169889820317877474760966, −9.351395122536100340317854363519, −8.743009354399470988109938801260, −7.56539711865008718892543032595, −6.86293069525421676075857030622, −5.89224758558411029740466682785, −5.42699359373176791888494613033, −3.98100791023143308855501196712, −2.86220030994256645266021020238, −1.69622047092740210548819898228, 0.62532817631652801605204159546, 1.11013987304757603036125882511, 2.73857273264401053546307640316, 3.90669438020426885397198553718, 4.69359914818304780008684774519, 5.59386237614709180812440199282, 6.87681013643337836745568727158, 7.87041032742478185698218687446, 8.591820916178147046662568962231, 9.721711641415808794454368451111

Graph of the ZZ-function along the critical line