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Properties

Label 2-810-3.2-c2-0-4
Degree 22
Conductor 810810
Sign −i-i
Analytic cond. 22.070922.0709
Root an. cond. 4.697964.69796
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.00·4-s + 2.23i·5-s − 1.06·7-s + 2.82i·8-s + 3.16·10-s + 0.743i·11-s − 3.34·13-s + 1.50i·14-s + 4.00·16-s − 27.5i·17-s + 6.98·19-s − 4.47i·20-s + 1.05·22-s + 31.9i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + 0.447i·5-s − 0.151·7-s + 0.353i·8-s + 0.316·10-s + 0.0675i·11-s − 0.257·13-s + 0.107i·14-s + 0.250·16-s − 1.62i·17-s + 0.367·19-s − 0.223i·20-s + 0.0477·22-s + 1.38i·23-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 810810    =    2⋅34⋅52 \cdot 3^{4} \cdot 5
Sign: −i-i
Analytic conductor: 22.070922.0709
Root analytic conductor: 4.697964.69796
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ810(161,⋅)\chi_{810} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 810, ( :1), −i)(2,\ 810,\ (\ :1),\ -i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.69816956430.6981695643
L(12)L(\frac12) ≈\approx 0.69816956430.6981695643
L(2)L(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+1.41iT 1 + 1.41iT
3 1 1
5 1−2.23iT 1 - 2.23iT
good7 1+1.06T+49T2 1 + 1.06T + 49T^{2}
11 1−0.743iT−121T2 1 - 0.743iT - 121T^{2}
13 1+3.34T+169T2 1 + 3.34T + 169T^{2}
17 1+27.5iT−289T2 1 + 27.5iT - 289T^{2}
19 1−6.98T+361T2 1 - 6.98T + 361T^{2}
23 1−31.9iT−529T2 1 - 31.9iT - 529T^{2}
29 1−48.9iT−841T2 1 - 48.9iT - 841T^{2}
31 1+47.6T+961T2 1 + 47.6T + 961T^{2}
37 1+31.0T+1.36e3T2 1 + 31.0T + 1.36e3T^{2}
41 1−3.95iT−1.68e3T2 1 - 3.95iT - 1.68e3T^{2}
43 1+2.93T+1.84e3T2 1 + 2.93T + 1.84e3T^{2}
47 1−67.9iT−2.20e3T2 1 - 67.9iT - 2.20e3T^{2}
53 1−40.2iT−2.80e3T2 1 - 40.2iT - 2.80e3T^{2}
59 1−7.63iT−3.48e3T2 1 - 7.63iT - 3.48e3T^{2}
61 1−66.5T+3.72e3T2 1 - 66.5T + 3.72e3T^{2}
67 1+23.1T+4.48e3T2 1 + 23.1T + 4.48e3T^{2}
71 1−103.iT−5.04e3T2 1 - 103. iT - 5.04e3T^{2}
73 1+127.T+5.32e3T2 1 + 127.T + 5.32e3T^{2}
79 1+53.8T+6.24e3T2 1 + 53.8T + 6.24e3T^{2}
83 1+96.2iT−6.88e3T2 1 + 96.2iT - 6.88e3T^{2}
89 1−54.5iT−7.92e3T2 1 - 54.5iT - 7.92e3T^{2}
97 1+40.4T+9.40e3T2 1 + 40.4T + 9.40e3T^{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

−10.28097368530386303067545986126, −9.476608200000072264816730627427, −8.914579927345182233383591434395, −7.53748659789227394002086942787, −7.04851346880608601947509879349, −5.64233635929329736015265911662, −4.87309549588299135139500778715, −3.55290748280356609515732830131, −2.81063948627980814993746525638, −1.44812738972558057628546339892, 0.23653998284348764922500713510, 1.93014570941731228284661198040, 3.58313084179476109013421820327, 4.49181874821930126654352968674, 5.53925284201178411811381627454, 6.29808893952676854191668881825, 7.24017263704706710115552641107, 8.240570722787287986929595453417, 8.726414845104979603020106180327, 9.785710856710900509289637132436

Graph of the ZZ-function along the critical line