↑

Properties

Label 2-21e2-63.4-c1-0-17
Degree 22
Conductor 441441
Sign 0.605+0.795i0.605 + 0.795i
Analytic cond. 3.521403.52140
Root an. cond. 1.876541.87654
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  + (0.439 − 0.761i)2-s + (−1.11 − 1.32i)3-s + (0.613 + 1.06i)4-s + 1.34·5-s + (−1.5 + 0.264i)6-s + 2.83·8-s + (−0.520 + 2.95i)9-s + (0.592 − 1.02i)10-s + 1.65·11-s + (0.726 − 1.99i)12-s + (1.68 − 2.91i)13-s + (−1.5 − 1.78i)15-s + (0.0209 − 0.0362i)16-s + (−0.233 + 0.405i)17-s + (2.02 + 1.69i)18-s + (1.61 + 2.79i)19-s + ⋯
L(s)  = 1  + (0.310 − 0.538i)2-s + (−0.642 − 0.766i)3-s + (0.306 + 0.531i)4-s + 0.602·5-s + (−0.612 + 0.107i)6-s + 1.00·8-s + (−0.173 + 0.984i)9-s + (0.187 − 0.324i)10-s + 0.498·11-s + (0.209 − 0.576i)12-s + (0.467 − 0.809i)13-s + (−0.387 − 0.461i)15-s + (0.00523 − 0.00906i)16-s + (−0.0567 + 0.0982i)17-s + (0.476 + 0.399i)18-s + (0.370 + 0.641i)19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.605+0.795i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.605+0.795i0.605 + 0.795i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(67,⋅)\chi_{441} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), 0.605+0.795i)(2,\ 441,\ (\ :1/2),\ 0.605 + 0.795i)

Particular Values

L(1)L(1) ≈\approx 1.54687−0.766828i1.54687 - 0.766828i
L(12)L(\frac12) ≈\approx 1.54687−0.766828i1.54687 - 0.766828i
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+(1.11+1.32i)T 1 + (1.11 + 1.32i)T
7 1 1
good2 1+(−0.439+0.761i)T+(−1−1.73i)T2 1 + (-0.439 + 0.761i)T + (-1 - 1.73i)T^{2}
5 1−1.34T+5T2 1 - 1.34T + 5T^{2}
11 1−1.65T+11T2 1 - 1.65T + 11T^{2}
13 1+(−1.68+2.91i)T+(−6.5−11.2i)T2 1 + (-1.68 + 2.91i)T + (-6.5 - 11.2i)T^{2}
17 1+(0.233−0.405i)T+(−8.5−14.7i)T2 1 + (0.233 - 0.405i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.61−2.79i)T+(−9.5+16.4i)T2 1 + (-1.61 - 2.79i)T + (-9.5 + 16.4i)T^{2}
23 1−8.94T+23T2 1 - 8.94T + 23T^{2}
29 1+(3.13+5.42i)T+(−14.5+25.1i)T2 1 + (3.13 + 5.42i)T + (-14.5 + 25.1i)T^{2}
31 1+(4.61+7.99i)T+(−15.5+26.8i)T2 1 + (4.61 + 7.99i)T + (-15.5 + 26.8i)T^{2}
37 1+(4.61+7.99i)T+(−18.5+32.0i)T2 1 + (4.61 + 7.99i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.70−2.95i)T+(−20.5−35.5i)T2 1 + (1.70 - 2.95i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.20−3.82i)T+(−21.5+37.2i)T2 1 + (-2.20 - 3.82i)T + (-21.5 + 37.2i)T^{2}
47 1+(4.67−8.10i)T+(−23.5−40.7i)T2 1 + (4.67 - 8.10i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.286+0.497i)T+(−26.5−45.8i)T2 1 + (-0.286 + 0.497i)T + (-26.5 - 45.8i)T^{2}
59 1+(−5.19−9.00i)T+(−29.5+51.0i)T2 1 + (-5.19 - 9.00i)T + (-29.5 + 51.0i)T^{2}
61 1+(3.81−6.61i)T+(−30.5−52.8i)T2 1 + (3.81 - 6.61i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.298+0.516i)T+(−33.5+58.0i)T2 1 + (0.298 + 0.516i)T + (-33.5 + 58.0i)T^{2}
71 1+0.554T+71T2 1 + 0.554T + 71T^{2}
73 1+(1.02−1.77i)T+(−36.5−63.2i)T2 1 + (1.02 - 1.77i)T + (-36.5 - 63.2i)T^{2}
79 1+(−1.20+2.08i)T+(−39.5−68.4i)T2 1 + (-1.20 + 2.08i)T + (-39.5 - 68.4i)T^{2}
83 1+(−7.52−13.0i)T+(−41.5+71.8i)T2 1 + (-7.52 - 13.0i)T + (-41.5 + 71.8i)T^{2}
89 1+(4.54+7.86i)T+(−44.5+77.0i)T2 1 + (4.54 + 7.86i)T + (-44.5 + 77.0i)T^{2}
97 1+(−0.949−1.64i)T+(−48.5+84.0i)T2 1 + (-0.949 - 1.64i)T + (-48.5 + 84.0i)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

−11.20483320511531534068090446810, −10.47206367295542940216042914310, −9.281272626305013244012971528598, −7.980938120989918386908790374740, −7.31625391447585698542672386689, −6.20360106806608713046304893021, −5.38436388529616352901938425534, −3.95165543361171249301758955541, −2.61316087231832447365979469722, −1.40000977447550774505957249211, 1.50792248904301098063192232970, 3.50526435110638626345219114257, 4.92223315931463880684090212539, 5.38230812697238564514380319898, 6.61834785531629225917214233497, 6.94858058933185393775940421850, 8.853756626599044319479436465524, 9.462992366450406661789618286963, 10.48549682056247329527793168971, 11.09418933271149358609166866000

Graph of the ZZ-function along the critical line