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Properties

Label 2-810-9.4-c3-0-26
Degree 22
Conductor 810810
Sign 0.342−0.939i0.342 - 0.939i
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  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (2.5 − 4.33i)5-s + (5.63 + 9.75i)7-s − 7.99·8-s + 10·10-s + (0.526 + 0.911i)11-s + (6.73 − 11.6i)13-s + (−11.2 + 19.5i)14-s + (−8 − 13.8i)16-s + 136.·17-s − 46.7·19-s + (10 + 17.3i)20-s + (−1.05 + 1.82i)22-s + (−10.9 + 18.9i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (0.304 + 0.526i)7-s − 0.353·8-s + 0.316·10-s + (0.0144 + 0.0249i)11-s + (0.143 − 0.248i)13-s + (−0.215 + 0.372i)14-s + (−0.125 − 0.216i)16-s + 1.95·17-s − 0.564·19-s + (0.111 + 0.193i)20-s + (−0.0102 + 0.0176i)22-s + (−0.0992 + 0.171i)23-s + ⋯

Functional equation

Λ(s)=(810s/2 ΓC(s) L(s)=((0.342−0.939i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(810s/2 ΓC(s+3/2) L(s)=((0.342−0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 810810    =    2⋅34⋅52 \cdot 3^{4} \cdot 5
Sign: 0.342−0.939i0.342 - 0.939i
Analytic conductor: 47.791547.7915
Root analytic conductor: 6.913146.91314
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ810(271,⋅)\chi_{810} (271, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 810, ( :3/2), 0.342−0.939i)(2,\ 810,\ (\ :3/2),\ 0.342 - 0.939i)

Particular Values

L(2)L(2) ≈\approx 2.7054619642.705461964
L(12)L(\frac12) ≈\approx 2.7054619642.705461964
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+(−1−1.73i)T 1 + (-1 - 1.73i)T
3 1 1
5 1+(−2.5+4.33i)T 1 + (-2.5 + 4.33i)T
good7 1+(−5.63−9.75i)T+(−171.5+297.i)T2 1 + (-5.63 - 9.75i)T + (-171.5 + 297. i)T^{2}
11 1+(−0.526−0.911i)T+(−665.5+1.15e3i)T2 1 + (-0.526 - 0.911i)T + (-665.5 + 1.15e3i)T^{2}
13 1+(−6.73+11.6i)T+(−1.09e3−1.90e3i)T2 1 + (-6.73 + 11.6i)T + (-1.09e3 - 1.90e3i)T^{2}
17 1−136.T+4.91e3T2 1 - 136.T + 4.91e3T^{2}
19 1+46.7T+6.85e3T2 1 + 46.7T + 6.85e3T^{2}
23 1+(10.9−18.9i)T+(−6.08e3−1.05e4i)T2 1 + (10.9 - 18.9i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(54.1+93.7i)T+(−1.21e4+2.11e4i)T2 1 + (54.1 + 93.7i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+(−28.5+49.4i)T+(−1.48e4−2.57e4i)T2 1 + (-28.5 + 49.4i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1−223.T+5.06e4T2 1 - 223.T + 5.06e4T^{2}
41 1+(34.4−59.6i)T+(−3.44e4−5.96e4i)T2 1 + (34.4 - 59.6i)T + (-3.44e4 - 5.96e4i)T^{2}
43 1+(−177.−307.i)T+(−3.97e4+6.88e4i)T2 1 + (-177. - 307. i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+(−241.−417.i)T+(−5.19e4+8.99e4i)T2 1 + (-241. - 417. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+110.T+1.48e5T2 1 + 110.T + 1.48e5T^{2}
59 1+(328.−569.i)T+(−1.02e5−1.77e5i)T2 1 + (328. - 569. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−19.2−33.3i)T+(−1.13e5+1.96e5i)T2 1 + (-19.2 - 33.3i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(−339.+587.i)T+(−1.50e5−2.60e5i)T2 1 + (-339. + 587. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1−572.T+3.57e5T2 1 - 572.T + 3.57e5T^{2}
73 1−107.T+3.89e5T2 1 - 107.T + 3.89e5T^{2}
79 1+(−51.8−89.8i)T+(−2.46e5+4.26e5i)T2 1 + (-51.8 - 89.8i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+(−472.−818.i)T+(−2.85e5+4.95e5i)T2 1 + (-472. - 818. i)T + (-2.85e5 + 4.95e5i)T^{2}
89 1−577.T+7.04e5T2 1 - 577.T + 7.04e5T^{2}
97 1+(−593.−1.02e3i)T+(−4.56e5+7.90e5i)T2 1 + (-593. - 1.02e3i)T + (-4.56e5 + 7.90e5i)T^{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

−9.797658348908456489723316657561, −9.146351126453074516938748310039, −8.031935986624637377054366089741, −7.69911736930723468538336659327, −6.27093864395003992146646451887, −5.69100925684587731457847474122, −4.82923205793568458789127031290, −3.75211657987601716501074565985, −2.53501693111668176642457709162, −1.03195142390989393358185957079, 0.803313199201397178641016285336, 1.97015919960354392415525750369, 3.21627577360421039985220404794, 4.04559945472064871810612492962, 5.17964717221696286301883367144, 6.02150774584011956563649394421, 7.09900721521468538457229585937, 7.961340776269900822644769749401, 9.020278646394158419741481294573, 9.981627042960495391791376225582

Graph of the ZZ-function along the critical line