↑

Properties

Label 2-735-49.22-c1-0-4
Degree 22
Conductor 735735
Sign −0.880−0.473i-0.880 - 0.473i
Analytic cond. 5.869005.86900
Root an. cond. 2.422602.42260
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  + (0.991 − 0.477i)2-s + (−0.222 + 0.974i)3-s + (−0.491 + 0.616i)4-s + (−0.222 + 0.974i)5-s + (0.244 + 1.07i)6-s + (−1.66 − 2.05i)7-s + (−0.682 + 2.99i)8-s + (−0.900 − 0.433i)9-s + (0.244 + 1.07i)10-s + (1.44 − 0.696i)11-s + (−0.491 − 0.616i)12-s + (−4.16 + 2.00i)13-s + (−2.63 − 1.23i)14-s + (−0.900 − 0.433i)15-s + (0.400 + 1.75i)16-s + (1.82 + 2.28i)17-s + ⋯
L(s)  = 1  + (0.701 − 0.337i)2-s + (−0.128 + 0.562i)3-s + (−0.245 + 0.308i)4-s + (−0.0995 + 0.436i)5-s + (0.0999 + 0.438i)6-s + (−0.631 − 0.775i)7-s + (−0.241 + 1.05i)8-s + (−0.300 − 0.144i)9-s + (0.0774 + 0.339i)10-s + (0.435 − 0.209i)11-s + (−0.141 − 0.178i)12-s + (−1.15 + 0.556i)13-s + (−0.704 − 0.330i)14-s + (−0.232 − 0.112i)15-s + (0.100 + 0.438i)16-s + (0.441 + 0.554i)17-s + ⋯

Functional equation

Λ(s)=(735s/2 ΓC(s) L(s)=((−0.880−0.473i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(735s/2 ΓC(s+1/2) L(s)=((−0.880−0.473i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    3⋅5⋅723 \cdot 5 \cdot 7^{2}
Sign: −0.880−0.473i-0.880 - 0.473i
Analytic conductor: 5.869005.86900
Root analytic conductor: 2.422602.42260
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ735(316,⋅)\chi_{735} (316, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 735, ( :1/2), −0.880−0.473i)(2,\ 735,\ (\ :1/2),\ -0.880 - 0.473i)

Particular Values

L(1)L(1) ≈\approx 0.197650+0.785059i0.197650 + 0.785059i
L(12)L(\frac12) ≈\approx 0.197650+0.785059i0.197650 + 0.785059i
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+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
5 1+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
7 1+(1.66+2.05i)T 1 + (1.66 + 2.05i)T
good2 1+(−0.991+0.477i)T+(1.24−1.56i)T2 1 + (-0.991 + 0.477i)T + (1.24 - 1.56i)T^{2}
11 1+(−1.44+0.696i)T+(6.85−8.60i)T2 1 + (-1.44 + 0.696i)T + (6.85 - 8.60i)T^{2}
13 1+(4.16−2.00i)T+(8.10−10.1i)T2 1 + (4.16 - 2.00i)T + (8.10 - 10.1i)T^{2}
17 1+(−1.82−2.28i)T+(−3.78+16.5i)T2 1 + (-1.82 - 2.28i)T + (-3.78 + 16.5i)T^{2}
19 1+3.21T+19T2 1 + 3.21T + 19T^{2}
23 1+(3.76−4.72i)T+(−5.11−22.4i)T2 1 + (3.76 - 4.72i)T + (-5.11 - 22.4i)T^{2}
29 1+(2.93+3.68i)T+(−6.45+28.2i)T2 1 + (2.93 + 3.68i)T + (-6.45 + 28.2i)T^{2}
31 1+8.97T+31T2 1 + 8.97T + 31T^{2}
37 1+(−5.34−6.69i)T+(−8.23+36.0i)T2 1 + (-5.34 - 6.69i)T + (-8.23 + 36.0i)T^{2}
41 1+(−1.41+6.20i)T+(−36.9−17.7i)T2 1 + (-1.41 + 6.20i)T + (-36.9 - 17.7i)T^{2}
43 1+(0.471+2.06i)T+(−38.7+18.6i)T2 1 + (0.471 + 2.06i)T + (-38.7 + 18.6i)T^{2}
47 1+(4.37−2.10i)T+(29.3−36.7i)T2 1 + (4.37 - 2.10i)T + (29.3 - 36.7i)T^{2}
53 1+(−1.69+2.12i)T+(−11.7−51.6i)T2 1 + (-1.69 + 2.12i)T + (-11.7 - 51.6i)T^{2}
59 1+(−1.83−8.05i)T+(−53.1+25.5i)T2 1 + (-1.83 - 8.05i)T + (-53.1 + 25.5i)T^{2}
61 1+(−1.27−1.60i)T+(−13.5+59.4i)T2 1 + (-1.27 - 1.60i)T + (-13.5 + 59.4i)T^{2}
67 1−15.1T+67T2 1 - 15.1T + 67T^{2}
71 1+(3.55−4.46i)T+(−15.7−69.2i)T2 1 + (3.55 - 4.46i)T + (-15.7 - 69.2i)T^{2}
73 1+(−13.3−6.42i)T+(45.5+57.0i)T2 1 + (-13.3 - 6.42i)T + (45.5 + 57.0i)T^{2}
79 1−13.6T+79T2 1 - 13.6T + 79T^{2}
83 1+(−3.65−1.76i)T+(51.7+64.8i)T2 1 + (-3.65 - 1.76i)T + (51.7 + 64.8i)T^{2}
89 1+(4.77+2.30i)T+(55.4+69.5i)T2 1 + (4.77 + 2.30i)T + (55.4 + 69.5i)T^{2}
97 1+3.55T+97T2 1 + 3.55T + 97T^{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

−10.87314001977976466611537819622, −9.892814240662213415441259520878, −9.313261398876373775472143405944, −8.113588938143048349259408416017, −7.23737217873728916735989976518, −6.16918893872649989636318098355, −5.15135586400547945304024512545, −3.93508328240834357184428958710, −3.71969122310073652312972329340, −2.32766914270782846278596804384, 0.31875035698296767602586071668, 2.18222025485305658700109295578, 3.56882238336160611824868964835, 4.79376278899184797471361619629, 5.52277681345496338773359377019, 6.32653180656338237949972466381, 7.16095766189503127191717892528, 8.221023381254401327551362121716, 9.339494912399552231736740745991, 9.733356275682150103350689692819

Graph of the ZZ-function along the critical line