↑

Properties

Label 2-810-45.29-c2-0-5
Degree 22
Conductor 810810
Sign 0.207−0.978i0.207 - 0.978i
Analytic cond. 22.070922.0709
Root an. cond. 4.697964.69796
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (−4.24 − 2.64i)5-s + (−11.8 − 6.84i)7-s + 2.82·8-s + (6.24 − 3.32i)10-s + (−10.6 − 6.15i)11-s + (−14.7 + 8.50i)13-s + (16.7 − 9.67i)14-s + (−2.00 + 3.46i)16-s + 6.89·17-s − 7.24·19-s + (−0.346 + 9.99i)20-s + (15.0 − 8.70i)22-s + (−17.3 − 30.1i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.848 − 0.529i)5-s + (−1.69 − 0.977i)7-s + 0.353·8-s + (0.624 − 0.332i)10-s + (−0.969 − 0.559i)11-s + (−1.13 + 0.654i)13-s + (1.19 − 0.691i)14-s + (−0.125 + 0.216i)16-s + 0.405·17-s − 0.381·19-s + (−0.0173 + 0.499i)20-s + (0.685 − 0.395i)22-s + (−0.756 − 1.31i)23-s + ⋯

Functional equation

Λ(s)=(810s/2 ΓC(s) L(s)=((0.207−0.978i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(810s/2 ΓC(s+1) L(s)=((0.207−0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 810810    =    2⋅34⋅52 \cdot 3^{4} \cdot 5
Sign: 0.207−0.978i0.207 - 0.978i
Analytic conductor: 22.070922.0709
Root analytic conductor: 4.697964.69796
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ810(269,⋅)\chi_{810} (269, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 810, ( :1), 0.207−0.978i)(2,\ 810,\ (\ :1),\ 0.207 - 0.978i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.24611221610.2461122161
L(12)L(\frac12) ≈\approx 0.24611221610.2461122161
L(2)L(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)
bad2 1+(0.707−1.22i)T 1 + (0.707 - 1.22i)T
3 1 1
5 1+(4.24+2.64i)T 1 + (4.24 + 2.64i)T
good7 1+(11.8+6.84i)T+(24.5+42.4i)T2 1 + (11.8 + 6.84i)T + (24.5 + 42.4i)T^{2}
11 1+(10.6+6.15i)T+(60.5+104.i)T2 1 + (10.6 + 6.15i)T + (60.5 + 104. i)T^{2}
13 1+(14.7−8.50i)T+(84.5−146.i)T2 1 + (14.7 - 8.50i)T + (84.5 - 146. i)T^{2}
17 1−6.89T+289T2 1 - 6.89T + 289T^{2}
19 1+7.24T+361T2 1 + 7.24T + 361T^{2}
23 1+(17.3+30.1i)T+(−264.5+458.i)T2 1 + (17.3 + 30.1i)T + (-264.5 + 458. i)T^{2}
29 1+(−18.3−10.5i)T+(420.5+728.i)T2 1 + (-18.3 - 10.5i)T + (420.5 + 728. i)T^{2}
31 1+(−19.1−33.0i)T+(−480.5+832.i)T2 1 + (-19.1 - 33.0i)T + (-480.5 + 832. i)T^{2}
37 1+21.5iT−1.36e3T2 1 + 21.5iT - 1.36e3T^{2}
41 1+(−31.4+18.1i)T+(840.5−1.45e3i)T2 1 + (-31.4 + 18.1i)T + (840.5 - 1.45e3i)T^{2}
43 1+(5.40+3.11i)T+(924.5+1.60e3i)T2 1 + (5.40 + 3.11i)T + (924.5 + 1.60e3i)T^{2}
47 1+(−20.1+34.8i)T+(−1.10e3−1.91e3i)T2 1 + (-20.1 + 34.8i)T + (-1.10e3 - 1.91e3i)T^{2}
53 1−38.2T+2.80e3T2 1 - 38.2T + 2.80e3T^{2}
59 1+(36.0−20.8i)T+(1.74e3−3.01e3i)T2 1 + (36.0 - 20.8i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(−7.52+13.0i)T+(−1.86e3−3.22e3i)T2 1 + (-7.52 + 13.0i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(111.−64.3i)T+(2.24e3−3.88e3i)T2 1 + (111. - 64.3i)T + (2.24e3 - 3.88e3i)T^{2}
71 1+104.iT−5.04e3T2 1 + 104. iT - 5.04e3T^{2}
73 1−2.11iT−5.32e3T2 1 - 2.11iT - 5.32e3T^{2}
79 1+(−22.0+38.1i)T+(−3.12e3−5.40e3i)T2 1 + (-22.0 + 38.1i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−27.5+47.6i)T+(−3.44e3−5.96e3i)T2 1 + (-27.5 + 47.6i)T + (-3.44e3 - 5.96e3i)T^{2}
89 1−68.1iT−7.92e3T2 1 - 68.1iT - 7.92e3T^{2}
97 1+(87.6+50.6i)T+(4.70e3+8.14e3i)T2 1 + (87.6 + 50.6i)T + (4.70e3 + 8.14e3i)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

−10.28956798494027366298111407299, −9.286196382897542205465696607985, −8.498341823528658141168720608510, −7.53999544229502545546858311238, −6.96085629642522510578021725670, −6.06630321402146009790274767316, −4.83245970543263564561203747742, −3.96991360697020507313288918671, −2.81393725788715210346190998928, −0.59573891358505706121519967929, 0.17087934155385081822813630987, 2.55523594474710831199619177002, 2.94399051906611577100944918549, 4.12334414334476920317549091466, 5.43851261271415473900831965462, 6.45275355638705832968510965928, 7.54092451314877124607053673261, 8.060892410297131555419314410956, 9.331659504257992193053940161893, 9.933369447760236578170390557726

Graph of the ZZ-function along the critical line