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Properties

Label 2-675-135.94-c1-0-14
Degree 22
Conductor 675675
Sign 0.866−0.498i0.866 - 0.498i
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  + (−1.32 + 0.233i)2-s + (0.300 − 1.70i)3-s + (−0.173 + 0.0632i)4-s + 2.33i·6-s + (−0.237 + 0.652i)7-s + (2.54 − 1.47i)8-s + (−2.81 − 1.02i)9-s + (−3.52 + 2.95i)11-s + (0.0555 + 0.315i)12-s + (1.39 + 0.245i)13-s + (0.162 − 0.921i)14-s + (−2.75 + 2.31i)16-s + (3.35 + 1.93i)17-s + (3.98 + 0.701i)18-s + (3.53 + 6.11i)19-s + ⋯
L(s)  = 1  + (−0.938 + 0.165i)2-s + (0.173 − 0.984i)3-s + (−0.0868 + 0.0316i)4-s + 0.952i·6-s + (−0.0897 + 0.246i)7-s + (0.901 − 0.520i)8-s + (−0.939 − 0.342i)9-s + (−1.06 + 0.890i)11-s + (0.0160 + 0.0909i)12-s + (0.385 + 0.0679i)13-s + (0.0434 − 0.246i)14-s + (−0.688 + 0.577i)16-s + (0.814 + 0.470i)17-s + (0.938 + 0.165i)18-s + (0.810 + 1.40i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.866−0.498i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((0.866−0.498i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.866−0.498i0.866 - 0.498i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(499,⋅)\chi_{675} (499, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 0.866−0.498i)(2,\ 675,\ (\ :1/2),\ 0.866 - 0.498i)

Particular Values

L(1)L(1) ≈\approx 0.695868+0.185796i0.695868 + 0.185796i
L(12)L(\frac12) ≈\approx 0.695868+0.185796i0.695868 + 0.185796i
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+(−0.300+1.70i)T 1 + (-0.300 + 1.70i)T
5 1 1
good2 1+(1.32−0.233i)T+(1.87−0.684i)T2 1 + (1.32 - 0.233i)T + (1.87 - 0.684i)T^{2}
7 1+(0.237−0.652i)T+(−5.36−4.49i)T2 1 + (0.237 - 0.652i)T + (-5.36 - 4.49i)T^{2}
11 1+(3.52−2.95i)T+(1.91−10.8i)T2 1 + (3.52 - 2.95i)T + (1.91 - 10.8i)T^{2}
13 1+(−1.39−0.245i)T+(12.2+4.44i)T2 1 + (-1.39 - 0.245i)T + (12.2 + 4.44i)T^{2}
17 1+(−3.35−1.93i)T+(8.5+14.7i)T2 1 + (-3.35 - 1.93i)T + (8.5 + 14.7i)T^{2}
19 1+(−3.53−6.11i)T+(−9.5+16.4i)T2 1 + (-3.53 - 6.11i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.30+3.59i)T+(−17.6+14.7i)T2 1 + (1.30 + 3.59i)T + (-17.6 + 14.7i)T^{2}
29 1+(0.851+4.82i)T+(−27.2+9.91i)T2 1 + (0.851 + 4.82i)T + (-27.2 + 9.91i)T^{2}
31 1+(−0.786+0.286i)T+(23.7−19.9i)T2 1 + (-0.786 + 0.286i)T + (23.7 - 19.9i)T^{2}
37 1+(−6.91−3.99i)T+(18.5+32.0i)T2 1 + (-6.91 - 3.99i)T + (18.5 + 32.0i)T^{2}
41 1+(1.36−7.74i)T+(−38.5−14.0i)T2 1 + (1.36 - 7.74i)T + (-38.5 - 14.0i)T^{2}
43 1+(−1.33−1.59i)T+(−7.46+42.3i)T2 1 + (-1.33 - 1.59i)T + (-7.46 + 42.3i)T^{2}
47 1+(−2.35+6.46i)T+(−36.0−30.2i)T2 1 + (-2.35 + 6.46i)T + (-36.0 - 30.2i)T^{2}
53 1+3.05iT−53T2 1 + 3.05iT - 53T^{2}
59 1+(−6.82−5.72i)T+(10.2+58.1i)T2 1 + (-6.82 - 5.72i)T + (10.2 + 58.1i)T^{2}
61 1+(−8.12−2.95i)T+(46.7+39.2i)T2 1 + (-8.12 - 2.95i)T + (46.7 + 39.2i)T^{2}
67 1+(−9.30−1.64i)T+(62.9+22.9i)T2 1 + (-9.30 - 1.64i)T + (62.9 + 22.9i)T^{2}
71 1+(−2.90+5.02i)T+(−35.5−61.4i)T2 1 + (-2.90 + 5.02i)T + (-35.5 - 61.4i)T^{2}
73 1+(4.68−2.70i)T+(36.5−63.2i)T2 1 + (4.68 - 2.70i)T + (36.5 - 63.2i)T^{2}
79 1+(−2.27−12.9i)T+(−74.2+27.0i)T2 1 + (-2.27 - 12.9i)T + (-74.2 + 27.0i)T^{2}
83 1+(1.11−0.197i)T+(77.9−28.3i)T2 1 + (1.11 - 0.197i)T + (77.9 - 28.3i)T^{2}
89 1+(−0.368−0.637i)T+(−44.5+77.0i)T2 1 + (-0.368 - 0.637i)T + (-44.5 + 77.0i)T^{2}
97 1+(5.36+6.39i)T+(−16.8+95.5i)T2 1 + (5.36 + 6.39i)T + (-16.8 + 95.5i)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

−10.12444590071057972626949905274, −9.807624564525901254251490813176, −8.547316996741246588766523143508, −7.978210143242003999672557735940, −7.46975732032247648366759491313, −6.35241060450892734967647982055, −5.35730637163232270355990421131, −3.91980607458642315216404537444, −2.45189780804244914999831040983, −1.14658906272257958365919149968, 0.62849322601942629842673597837, 2.66085796499277951586985337821, 3.74744256377714989157463753481, 5.06748380628653155979175666852, 5.58184593789101889754122754338, 7.31869722782587151999877634616, 8.086160729585693659491367173085, 8.928661338589449477849925071449, 9.522426375888190694028423593847, 10.32528780555011337614720903119

Graph of the ZZ-function along the critical line