↑

Properties

Label 2-675-135.94-c1-0-6
Degree 22
Conductor 675675
Sign −0.918−0.394i-0.918 - 0.394i
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

Related objects

Downloads

Learn more

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)=((−0.918−0.394i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((−0.918−0.394i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: −0.918−0.394i-0.918 - 0.394i
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), −0.918−0.394i)(2,\ 675,\ (\ :1/2),\ -0.918 - 0.394i)

Particular Values

L(1)L(1) ≈\approx 0.207111+1.00755i0.207111 + 1.00755i
L(12)L(\frac12) ≈\approx 0.207111+1.00755i0.207111 + 1.00755i
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}
show more
show less
   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

−10.94040078131962786884396309620, −9.984557415827740361558359387009, −9.438294498897692992046778941504, −8.326887502639630590581186986575, −7.37544855054741443987045256808, −5.92333869473176599986703277849, −5.17139307978053558811488085590, −4.49404417559745415905614292010, −3.57610653303171274586826565959, −2.50679298021494174889355704822, 0.38994917036326733227311862101, 2.40108414581629192260220702580, 3.38169893691561332542805514666, 5.00650865370543436142862907868, 5.36880619298302802133444556773, 6.51347444925012189376122919599, 7.14695504223167205053281656194, 8.463647418398124602399870967266, 8.909032932960880472626132263464, 10.32667926502934010014354355638

Graph of the ZZ-function along the critical line