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Properties

Label 2-675-27.16-c1-0-25
Degree 22
Conductor 675675
Sign 0.195+0.980i0.195 + 0.980i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.194 + 0.163i)2-s + (−1.72 − 0.154i)3-s + (−0.336 − 1.90i)4-s + (−0.310 − 0.312i)6-s + (−0.449 + 2.55i)7-s + (0.500 − 0.866i)8-s + (2.95 + 0.534i)9-s + (2.07 − 0.753i)11-s + (0.284 + 3.34i)12-s + (1.11 − 0.932i)13-s + (−0.504 + 0.423i)14-s + (−3.39 + 1.23i)16-s + (1.17 + 2.04i)17-s + (0.487 + 0.586i)18-s + (2.22 − 3.84i)19-s + ⋯
L(s)  = 1  + (0.137 + 0.115i)2-s + (−0.995 − 0.0894i)3-s + (−0.168 − 0.952i)4-s + (−0.126 − 0.127i)6-s + (−0.170 + 0.964i)7-s + (0.176 − 0.306i)8-s + (0.983 + 0.178i)9-s + (0.624 − 0.227i)11-s + (0.0821 + 0.964i)12-s + (0.308 − 0.258i)13-s + (−0.134 + 0.113i)14-s + (−0.849 + 0.309i)16-s + (0.285 + 0.494i)17-s + (0.114 + 0.138i)18-s + (0.509 − 0.882i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.195+0.980i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((0.195+0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.195+0.980i0.195 + 0.980i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(151,⋅)\chi_{675} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 0.195+0.980i)(2,\ 675,\ (\ :1/2),\ 0.195 + 0.980i)

Particular Values

L(1)L(1) ≈\approx 0.784053−0.643238i0.784053 - 0.643238i
L(12)L(\frac12) ≈\approx 0.784053−0.643238i0.784053 - 0.643238i
L(32)L(\frac{3}{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)
bad3 1+(1.72+0.154i)T 1 + (1.72 + 0.154i)T
5 1 1
good2 1+(−0.194−0.163i)T+(0.347+1.96i)T2 1 + (-0.194 - 0.163i)T + (0.347 + 1.96i)T^{2}
7 1+(0.449−2.55i)T+(−6.57−2.39i)T2 1 + (0.449 - 2.55i)T + (-6.57 - 2.39i)T^{2}
11 1+(−2.07+0.753i)T+(8.42−7.07i)T2 1 + (-2.07 + 0.753i)T + (8.42 - 7.07i)T^{2}
13 1+(−1.11+0.932i)T+(2.25−12.8i)T2 1 + (-1.11 + 0.932i)T + (2.25 - 12.8i)T^{2}
17 1+(−1.17−2.04i)T+(−8.5+14.7i)T2 1 + (-1.17 - 2.04i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.22+3.84i)T+(−9.5−16.4i)T2 1 + (-2.22 + 3.84i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.22+6.94i)T+(−21.6+7.86i)T2 1 + (1.22 + 6.94i)T + (-21.6 + 7.86i)T^{2}
29 1+(4.88+4.10i)T+(5.03+28.5i)T2 1 + (4.88 + 4.10i)T + (5.03 + 28.5i)T^{2}
31 1+(1.13+6.45i)T+(−29.1+10.6i)T2 1 + (1.13 + 6.45i)T + (-29.1 + 10.6i)T^{2}
37 1+(−2.27−3.93i)T+(−18.5+32.0i)T2 1 + (-2.27 - 3.93i)T + (-18.5 + 32.0i)T^{2}
41 1+(−5.08+4.26i)T+(7.11−40.3i)T2 1 + (-5.08 + 4.26i)T + (7.11 - 40.3i)T^{2}
43 1+(−5.79+2.10i)T+(32.9−27.6i)T2 1 + (-5.79 + 2.10i)T + (32.9 - 27.6i)T^{2}
47 1+(−1.65+9.40i)T+(−44.1−16.0i)T2 1 + (-1.65 + 9.40i)T + (-44.1 - 16.0i)T^{2}
53 1+13.3T+53T2 1 + 13.3T + 53T^{2}
59 1+(−3.10−1.12i)T+(45.1+37.9i)T2 1 + (-3.10 - 1.12i)T + (45.1 + 37.9i)T^{2}
61 1+(1.57−8.95i)T+(−57.3−20.8i)T2 1 + (1.57 - 8.95i)T + (-57.3 - 20.8i)T^{2}
67 1+(−4.09+3.43i)T+(11.6−65.9i)T2 1 + (-4.09 + 3.43i)T + (11.6 - 65.9i)T^{2}
71 1+(1.67+2.89i)T+(−35.5+61.4i)T2 1 + (1.67 + 2.89i)T + (-35.5 + 61.4i)T^{2}
73 1+(−2.55+4.41i)T+(−36.5−63.2i)T2 1 + (-2.55 + 4.41i)T + (-36.5 - 63.2i)T^{2}
79 1+(−3.14−2.63i)T+(13.7+77.7i)T2 1 + (-3.14 - 2.63i)T + (13.7 + 77.7i)T^{2}
83 1+(2.65+2.22i)T+(14.4+81.7i)T2 1 + (2.65 + 2.22i)T + (14.4 + 81.7i)T^{2}
89 1+(−7.60+13.1i)T+(−44.5−77.0i)T2 1 + (-7.60 + 13.1i)T + (-44.5 - 77.0i)T^{2}
97 1+(−3.94+1.43i)T+(74.3−62.3i)T2 1 + (-3.94 + 1.43i)T + (74.3 - 62.3i)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

−10.39744918934205483813671800079, −9.530503556900063064645984536869, −8.814803871909850568426197865957, −7.47015764051502212671678865698, −6.24976474902157164988713953335, −5.99329903209469635659191615209, −5.04329720934670292787636013668, −4.03921045513942661902981737085, −2.17601497153984475711328464633, −0.66083246601424595510569048517, 1.37589929884699535774890583150, 3.44785414113164787540657396006, 4.08812186833448427814192756151, 5.12769052418870091021031391398, 6.26304322549355239681786026948, 7.30846829571090804571771653273, 7.69355779810173534322414381543, 9.208494277818809926620682769788, 9.803406616298174847908309718800, 11.02104816795485390705964086112

Graph of the ZZ-function along the critical line