↑

Properties

Label 2-675-27.16-c1-0-8
Degree 22
Conductor 675675
Sign 0.992−0.124i0.992 - 0.124i
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  + (−1.01 − 0.852i)2-s + (0.196 − 1.72i)3-s + (−0.0418 − 0.237i)4-s + (−1.66 + 1.58i)6-s + (−0.769 + 4.36i)7-s + (−1.48 + 2.57i)8-s + (−2.92 − 0.674i)9-s + (3.91 − 1.42i)11-s + (−0.416 + 0.0254i)12-s + (−1.15 + 0.972i)13-s + (4.50 − 3.77i)14-s + (3.25 − 1.18i)16-s + (0.568 + 0.985i)17-s + (2.39 + 3.17i)18-s + (−3.37 + 5.84i)19-s + ⋯
L(s)  = 1  + (−0.718 − 0.602i)2-s + (0.113 − 0.993i)3-s + (−0.0209 − 0.118i)4-s + (−0.680 + 0.645i)6-s + (−0.291 + 1.65i)7-s + (−0.525 + 0.910i)8-s + (−0.974 − 0.224i)9-s + (1.17 − 0.429i)11-s + (−0.120 + 0.00735i)12-s + (−0.321 + 0.269i)13-s + (1.20 − 1.01i)14-s + (0.812 − 0.295i)16-s + (0.137 + 0.238i)17-s + (0.564 + 0.748i)18-s + (−0.774 + 1.34i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.992−0.124i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((0.992−0.124i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.992−0.124i0.992 - 0.124i
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), 0.992−0.124i)(2,\ 675,\ (\ :1/2),\ 0.992 - 0.124i)

Particular Values

L(1)L(1) ≈\approx 0.727102+0.0455687i0.727102 + 0.0455687i
L(12)L(\frac12) ≈\approx 0.727102+0.0455687i0.727102 + 0.0455687i
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+(−0.196+1.72i)T 1 + (-0.196 + 1.72i)T
5 1 1
good2 1+(1.01+0.852i)T+(0.347+1.96i)T2 1 + (1.01 + 0.852i)T + (0.347 + 1.96i)T^{2}
7 1+(0.769−4.36i)T+(−6.57−2.39i)T2 1 + (0.769 - 4.36i)T + (-6.57 - 2.39i)T^{2}
11 1+(−3.91+1.42i)T+(8.42−7.07i)T2 1 + (-3.91 + 1.42i)T + (8.42 - 7.07i)T^{2}
13 1+(1.15−0.972i)T+(2.25−12.8i)T2 1 + (1.15 - 0.972i)T + (2.25 - 12.8i)T^{2}
17 1+(−0.568−0.985i)T+(−8.5+14.7i)T2 1 + (-0.568 - 0.985i)T + (-8.5 + 14.7i)T^{2}
19 1+(3.37−5.84i)T+(−9.5−16.4i)T2 1 + (3.37 - 5.84i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.38−7.86i)T+(−21.6+7.86i)T2 1 + (-1.38 - 7.86i)T + (-21.6 + 7.86i)T^{2}
29 1+(−1.89−1.58i)T+(5.03+28.5i)T2 1 + (-1.89 - 1.58i)T + (5.03 + 28.5i)T^{2}
31 1+(−0.369−2.09i)T+(−29.1+10.6i)T2 1 + (-0.369 - 2.09i)T + (-29.1 + 10.6i)T^{2}
37 1+(4.99+8.65i)T+(−18.5+32.0i)T2 1 + (4.99 + 8.65i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.384−0.322i)T+(7.11−40.3i)T2 1 + (0.384 - 0.322i)T + (7.11 - 40.3i)T^{2}
43 1+(0.102−0.0373i)T+(32.9−27.6i)T2 1 + (0.102 - 0.0373i)T + (32.9 - 27.6i)T^{2}
47 1+(−1.31+7.45i)T+(−44.1−16.0i)T2 1 + (-1.31 + 7.45i)T + (-44.1 - 16.0i)T^{2}
53 1−3.96T+53T2 1 - 3.96T + 53T^{2}
59 1+(−7.67−2.79i)T+(45.1+37.9i)T2 1 + (-7.67 - 2.79i)T + (45.1 + 37.9i)T^{2}
61 1+(2.14−12.1i)T+(−57.3−20.8i)T2 1 + (2.14 - 12.1i)T + (-57.3 - 20.8i)T^{2}
67 1+(2.25−1.88i)T+(11.6−65.9i)T2 1 + (2.25 - 1.88i)T + (11.6 - 65.9i)T^{2}
71 1+(3.78+6.56i)T+(−35.5+61.4i)T2 1 + (3.78 + 6.56i)T + (-35.5 + 61.4i)T^{2}
73 1+(5.12−8.88i)T+(−36.5−63.2i)T2 1 + (5.12 - 8.88i)T + (-36.5 - 63.2i)T^{2}
79 1+(−3.74−3.14i)T+(13.7+77.7i)T2 1 + (-3.74 - 3.14i)T + (13.7 + 77.7i)T^{2}
83 1+(2.78+2.34i)T+(14.4+81.7i)T2 1 + (2.78 + 2.34i)T + (14.4 + 81.7i)T^{2}
89 1+(−5.74+9.95i)T+(−44.5−77.0i)T2 1 + (-5.74 + 9.95i)T + (-44.5 - 77.0i)T^{2}
97 1+(7.36−2.68i)T+(74.3−62.3i)T2 1 + (7.36 - 2.68i)T + (74.3 - 62.3i)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

−10.48921700559530753948990031020, −9.398769314908662327968015619874, −8.857952632272465125490908783422, −8.327005366724167791213020037536, −7.02133684116257462502388942758, −5.89872442982806987612776979573, −5.57816963142236041174722253522, −3.48331344971189240677768288235, −2.27088256121941055234924192139, −1.45694438158152312893269646198, 0.50736953889802604768916836579, 3.01651975608031461906295509900, 4.11095164313952712896450314384, 4.65043299326080376589851286177, 6.50261907432109747227787523331, 6.89863012216865603043214078898, 7.987303993594420521245335860476, 8.816971863171235374917458027389, 9.579006712196541704690545239189, 10.22742481391169073451398159729

Graph of the ZZ-function along the critical line