↑

Properties

Label 2-666-111.14-c1-0-3
Degree 22
Conductor 666666
Sign 0.690+0.723i0.690 + 0.723i
Analytic cond. 5.318035.31803
Root an. cond. 2.306082.30608
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.448 + 1.67i)5-s + (−1 − 1.73i)7-s + (0.707 + 0.707i)8-s + 1.73·10-s + 3.86·11-s + (−4.09 − 1.09i)13-s + (−1.41 + 1.41i)14-s + (0.500 − 0.866i)16-s + (5.53 − 1.48i)17-s + (1 + 0.267i)19-s + (−0.448 − 1.67i)20-s + (−0.999 − 3.73i)22-s + (3.86 + 3.86i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.200 + 0.748i)5-s + (−0.377 − 0.654i)7-s + (0.249 + 0.249i)8-s + 0.547·10-s + 1.16·11-s + (−1.13 − 0.304i)13-s + (−0.377 + 0.377i)14-s + (0.125 − 0.216i)16-s + (1.34 − 0.359i)17-s + (0.229 + 0.0614i)19-s + (−0.100 − 0.374i)20-s + (−0.213 − 0.795i)22-s + (0.805 + 0.805i)23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((0.690+0.723i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((0.690+0.723i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: 0.690+0.723i0.690 + 0.723i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(125,⋅)\chi_{666} (125, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), 0.690+0.723i)(2,\ 666,\ (\ :1/2),\ 0.690 + 0.723i)

Particular Values

L(1)L(1) ≈\approx 1.16989−0.500493i1.16989 - 0.500493i
L(12)L(\frac12) ≈\approx 1.16989−0.500493i1.16989 - 0.500493i
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)
bad2 1+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
3 1 1
37 1+(−2.59+5.5i)T 1 + (-2.59 + 5.5i)T
good5 1+(0.448−1.67i)T+(−4.33−2.5i)T2 1 + (0.448 - 1.67i)T + (-4.33 - 2.5i)T^{2}
7 1+(1+1.73i)T+(−3.5+6.06i)T2 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2}
11 1−3.86T+11T2 1 - 3.86T + 11T^{2}
13 1+(4.09+1.09i)T+(11.2+6.5i)T2 1 + (4.09 + 1.09i)T + (11.2 + 6.5i)T^{2}
17 1+(−5.53+1.48i)T+(14.7−8.5i)T2 1 + (-5.53 + 1.48i)T + (14.7 - 8.5i)T^{2}
19 1+(−1−0.267i)T+(16.4+9.5i)T2 1 + (-1 - 0.267i)T + (16.4 + 9.5i)T^{2}
23 1+(−3.86−3.86i)T+23iT2 1 + (-3.86 - 3.86i)T + 23iT^{2}
29 1+(−6.50+6.50i)T−29iT2 1 + (-6.50 + 6.50i)T - 29iT^{2}
31 1+(−1.26−1.26i)T+31iT2 1 + (-1.26 - 1.26i)T + 31iT^{2}
41 1+(−0.258−0.448i)T+(−20.5+35.5i)T2 1 + (-0.258 - 0.448i)T + (-20.5 + 35.5i)T^{2}
43 1+(−4.73+4.73i)T−43iT2 1 + (-4.73 + 4.73i)T - 43iT^{2}
47 1−5.93iT−47T2 1 - 5.93iT - 47T^{2}
53 1+(−1.22−0.707i)T+(26.5+45.8i)T2 1 + (-1.22 - 0.707i)T + (26.5 + 45.8i)T^{2}
59 1+(9.14−2.44i)T+(51.0−29.5i)T2 1 + (9.14 - 2.44i)T + (51.0 - 29.5i)T^{2}
61 1+(2.42−9.06i)T+(−52.8−30.5i)T2 1 + (2.42 - 9.06i)T + (-52.8 - 30.5i)T^{2}
67 1+(−9+5.19i)T+(33.5−58.0i)T2 1 + (-9 + 5.19i)T + (33.5 - 58.0i)T^{2}
71 1+(7.58−4.38i)T+(35.5−61.4i)T2 1 + (7.58 - 4.38i)T + (35.5 - 61.4i)T^{2}
73 1+4iT−73T2 1 + 4iT - 73T^{2}
79 1+(13.9+3.73i)T+(68.4+39.5i)T2 1 + (13.9 + 3.73i)T + (68.4 + 39.5i)T^{2}
83 1+(4.89+2.82i)T+(41.5+71.8i)T2 1 + (4.89 + 2.82i)T + (41.5 + 71.8i)T^{2}
89 1+(−0.258−0.965i)T+(−77.0+44.5i)T2 1 + (-0.258 - 0.965i)T + (-77.0 + 44.5i)T^{2}
97 1+(−3.63+3.63i)T−97iT2 1 + (-3.63 + 3.63i)T - 97iT^{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

−10.28020140430667721634388690958, −9.807198582230523127516659484554, −8.941541239992399894596607486875, −7.55147404740296816803573963378, −7.18852166355302504907081719082, −5.93760924257146085989573322573, −4.62220237102513721150247863720, −3.55054841379858594156274084274, −2.75362200415880039695538384352, −0.989498649154790071725286313333, 1.12411464036829486731155354282, 2.98911111113836348909812297716, 4.42877417764614220942241956939, 5.17515251131172490298000152759, 6.26414451101063990667984611693, 7.03663564934327852840248745824, 8.119107045194168839773980726905, 8.889177597864080382002626113214, 9.495840932497176271964077507121, 10.34233733678853363130670933125

Graph of the ZZ-function along the critical line