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Properties

Label 2-675-135.113-c1-0-21
Degree 22
Conductor 675675
Sign 0.239+0.970i0.239 + 0.970i
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  + (−0.0777 − 0.00680i)2-s + (−1.54 − 0.788i)3-s + (−1.96 − 0.346i)4-s + (0.114 + 0.0718i)6-s + (1.14 + 0.804i)7-s + (0.301 + 0.0807i)8-s + (1.75 + 2.43i)9-s + (−1.25 + 3.43i)11-s + (2.75 + 2.08i)12-s + (−0.326 − 3.73i)13-s + (−0.0838 − 0.0703i)14-s + (3.72 + 1.35i)16-s + (4.17 − 1.11i)17-s + (−0.120 − 0.201i)18-s + (−1.40 + 0.808i)19-s + ⋯
L(s)  = 1  + (−0.0549 − 0.00481i)2-s + (−0.890 − 0.455i)3-s + (−0.981 − 0.173i)4-s + (0.0467 + 0.0293i)6-s + (0.434 + 0.304i)7-s + (0.106 + 0.0285i)8-s + (0.585 + 0.810i)9-s + (−0.377 + 1.03i)11-s + (0.795 + 0.600i)12-s + (−0.0906 − 1.03i)13-s + (−0.0224 − 0.0188i)14-s + (0.931 + 0.338i)16-s + (1.01 − 0.271i)17-s + (−0.0283 − 0.0473i)18-s + (−0.321 + 0.185i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.239+0.970i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((0.239+0.970i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.239+0.970i0.239 + 0.970i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(518,⋅)\chi_{675} (518, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 0.239+0.970i)(2,\ 675,\ (\ :1/2),\ 0.239 + 0.970i)

Particular Values

L(1)L(1) ≈\approx 0.579707−0.454184i0.579707 - 0.454184i
L(12)L(\frac12) ≈\approx 0.579707−0.454184i0.579707 - 0.454184i
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+(1.54+0.788i)T 1 + (1.54 + 0.788i)T
5 1 1
good2 1+(0.0777+0.00680i)T+(1.96+0.347i)T2 1 + (0.0777 + 0.00680i)T + (1.96 + 0.347i)T^{2}
7 1+(−1.14−0.804i)T+(2.39+6.57i)T2 1 + (-1.14 - 0.804i)T + (2.39 + 6.57i)T^{2}
11 1+(1.25−3.43i)T+(−8.42−7.07i)T2 1 + (1.25 - 3.43i)T + (-8.42 - 7.07i)T^{2}
13 1+(0.326+3.73i)T+(−12.8+2.25i)T2 1 + (0.326 + 3.73i)T + (-12.8 + 2.25i)T^{2}
17 1+(−4.17+1.11i)T+(14.7−8.5i)T2 1 + (-4.17 + 1.11i)T + (14.7 - 8.5i)T^{2}
19 1+(1.40−0.808i)T+(9.5−16.4i)T2 1 + (1.40 - 0.808i)T + (9.5 - 16.4i)T^{2}
23 1+(3.27+4.67i)T+(−7.86+21.6i)T2 1 + (3.27 + 4.67i)T + (-7.86 + 21.6i)T^{2}
29 1+(−3.26+2.73i)T+(5.03−28.5i)T2 1 + (-3.26 + 2.73i)T + (5.03 - 28.5i)T^{2}
31 1+(−0.336+1.91i)T+(−29.1−10.6i)T2 1 + (-0.336 + 1.91i)T + (-29.1 - 10.6i)T^{2}
37 1+(2.07+7.76i)T+(−32.0+18.5i)T2 1 + (2.07 + 7.76i)T + (-32.0 + 18.5i)T^{2}
41 1+(2.08−2.48i)T+(−7.11−40.3i)T2 1 + (2.08 - 2.48i)T + (-7.11 - 40.3i)T^{2}
43 1+(−0.347−0.746i)T+(−27.6+32.9i)T2 1 + (-0.347 - 0.746i)T + (-27.6 + 32.9i)T^{2}
47 1+(−0.661+0.945i)T+(−16.0−44.1i)T2 1 + (-0.661 + 0.945i)T + (-16.0 - 44.1i)T^{2}
53 1+(−7.50+7.50i)T−53iT2 1 + (-7.50 + 7.50i)T - 53iT^{2}
59 1+(−9.85+3.58i)T+(45.1−37.9i)T2 1 + (-9.85 + 3.58i)T + (45.1 - 37.9i)T^{2}
61 1+(1.72+9.76i)T+(−57.3+20.8i)T2 1 + (1.72 + 9.76i)T + (-57.3 + 20.8i)T^{2}
67 1+(−14.7+1.28i)T+(65.9−11.6i)T2 1 + (-14.7 + 1.28i)T + (65.9 - 11.6i)T^{2}
71 1+(−8.12−4.69i)T+(35.5+61.4i)T2 1 + (-8.12 - 4.69i)T + (35.5 + 61.4i)T^{2}
73 1+(0.0713−0.266i)T+(−63.2−36.5i)T2 1 + (0.0713 - 0.266i)T + (-63.2 - 36.5i)T^{2}
79 1+(7.24+8.63i)T+(−13.7+77.7i)T2 1 + (7.24 + 8.63i)T + (-13.7 + 77.7i)T^{2}
83 1+(1.48−17.0i)T+(−81.7−14.4i)T2 1 + (1.48 - 17.0i)T + (-81.7 - 14.4i)T^{2}
89 1+(−6.77−11.7i)T+(−44.5+77.0i)T2 1 + (-6.77 - 11.7i)T + (-44.5 + 77.0i)T^{2}
97 1+(12.5−5.86i)T+(62.3−74.3i)T2 1 + (12.5 - 5.86i)T + (62.3 - 74.3i)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

−10.12741143026469135043292850053, −9.834415578569730590264734384831, −8.295846073789354472611533442998, −7.893967903519704512625927035438, −6.72065007079807397105850901226, −5.49954035791701605145569680303, −5.10817658082820341761016621518, −4.01245521322469905794817801710, −2.17397414583837433014630913811, −0.57665717810707641534108718723, 1.12037495610579000663777122904, 3.42755179243139080965063520323, 4.29691131415629793427992537270, 5.19182162308766752335894843316, 5.95735551080234279128041818696, 7.16230689369667475375150585111, 8.233141465649350428823582934399, 8.997555539111953786399193226558, 9.965485626958989713048302001498, 10.54209242583094556517807107822

Graph of the ZZ-function along the critical line