↑

Properties

Label 2-209-19.7-c1-0-14
Degree 22
Conductor 209209
Sign 0.242+0.970i0.242 + 0.970i
Analytic cond. 1.668871.66887
Root an. cond. 1.291841.29184
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  + (1.00 + 1.74i)2-s + (−1.29 − 2.23i)3-s + (−1.02 + 1.77i)4-s + (−1.77 − 3.07i)5-s + (2.59 − 4.50i)6-s − 4.09·7-s − 0.111·8-s + (−1.82 + 3.16i)9-s + (3.57 − 6.19i)10-s − 11-s + 5.30·12-s + (2.55 − 4.42i)13-s + (−4.12 − 7.14i)14-s + (−4.58 + 7.93i)15-s + (1.94 + 3.36i)16-s + (0.559 + 0.969i)17-s + ⋯
L(s)  = 1  + (0.711 + 1.23i)2-s + (−0.744 − 1.29i)3-s + (−0.513 + 0.889i)4-s + (−0.794 − 1.37i)5-s + (1.06 − 1.83i)6-s − 1.54·7-s − 0.0393·8-s + (−0.609 + 1.05i)9-s + (1.13 − 1.95i)10-s − 0.301·11-s + 1.53·12-s + (0.708 − 1.22i)13-s + (−1.10 − 1.91i)14-s + (−1.18 + 2.04i)15-s + (0.485 + 0.841i)16-s + (0.135 + 0.235i)17-s + ⋯

Functional equation

Λ(s)=(209s/2 ΓC(s) L(s)=((0.242+0.970i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(209s/2 ΓC(s+1/2) L(s)=((0.242+0.970i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 209209    =    11⋅1911 \cdot 19
Sign: 0.242+0.970i0.242 + 0.970i
Analytic conductor: 1.668871.66887
Root analytic conductor: 1.291841.29184
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ209(45,⋅)\chi_{209} (45, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 209, ( :1/2), 0.242+0.970i)(2,\ 209,\ (\ :1/2),\ 0.242 + 0.970i)

Particular Values

L(1)L(1) ≈\approx 0.711288−0.555598i0.711288 - 0.555598i
L(12)L(\frac12) ≈\approx 0.711288−0.555598i0.711288 - 0.555598i
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)
bad11 1+T 1 + T
19 1+(−4.34−0.288i)T 1 + (-4.34 - 0.288i)T
good2 1+(−1.00−1.74i)T+(−1+1.73i)T2 1 + (-1.00 - 1.74i)T + (-1 + 1.73i)T^{2}
3 1+(1.29+2.23i)T+(−1.5+2.59i)T2 1 + (1.29 + 2.23i)T + (-1.5 + 2.59i)T^{2}
5 1+(1.77+3.07i)T+(−2.5+4.33i)T2 1 + (1.77 + 3.07i)T + (-2.5 + 4.33i)T^{2}
7 1+4.09T+7T2 1 + 4.09T + 7T^{2}
13 1+(−2.55+4.42i)T+(−6.5−11.2i)T2 1 + (-2.55 + 4.42i)T + (-6.5 - 11.2i)T^{2}
17 1+(−0.559−0.969i)T+(−8.5+14.7i)T2 1 + (-0.559 - 0.969i)T + (-8.5 + 14.7i)T^{2}
23 1+(−0.629+1.09i)T+(−11.5−19.9i)T2 1 + (-0.629 + 1.09i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3.20+5.55i)T+(−14.5−25.1i)T2 1 + (-3.20 + 5.55i)T + (-14.5 - 25.1i)T^{2}
31 1+0.563T+31T2 1 + 0.563T + 31T^{2}
37 1+1.00T+37T2 1 + 1.00T + 37T^{2}
41 1+(2.90+5.02i)T+(−20.5+35.5i)T2 1 + (2.90 + 5.02i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.45+5.98i)T+(−21.5+37.2i)T2 1 + (3.45 + 5.98i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.17+2.03i)T+(−23.5−40.7i)T2 1 + (-1.17 + 2.03i)T + (-23.5 - 40.7i)T^{2}
53 1+(−6.48+11.2i)T+(−26.5−45.8i)T2 1 + (-6.48 + 11.2i)T + (-26.5 - 45.8i)T^{2}
59 1+(−3.10−5.37i)T+(−29.5+51.0i)T2 1 + (-3.10 - 5.37i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.68−4.64i)T+(−30.5−52.8i)T2 1 + (2.68 - 4.64i)T + (-30.5 - 52.8i)T^{2}
67 1+(5.01−8.69i)T+(−33.5−58.0i)T2 1 + (5.01 - 8.69i)T + (-33.5 - 58.0i)T^{2}
71 1+(0.778+1.34i)T+(−35.5+61.4i)T2 1 + (0.778 + 1.34i)T + (-35.5 + 61.4i)T^{2}
73 1+(3.65+6.32i)T+(−36.5+63.2i)T2 1 + (3.65 + 6.32i)T + (-36.5 + 63.2i)T^{2}
79 1+(−7.30−12.6i)T+(−39.5+68.4i)T2 1 + (-7.30 - 12.6i)T + (-39.5 + 68.4i)T^{2}
83 1−3.84T+83T2 1 - 3.84T + 83T^{2}
89 1+(−0.742+1.28i)T+(−44.5−77.0i)T2 1 + (-0.742 + 1.28i)T + (-44.5 - 77.0i)T^{2}
97 1+(−8.33−14.4i)T+(−48.5+84.0i)T2 1 + (-8.33 - 14.4i)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

−12.50297726101297305571511755911, −11.87483760312201092792962192598, −10.25640477507787364075172487618, −8.615146900706273945723489935305, −7.77407574578923115205169856717, −6.92075366123956759879878028819, −5.91172210847437745618766481465, −5.28220824918710337904748437548, −3.69711655077345180306563399532, −0.68879155200787775693656748897, 3.11481419293209081964884584851, 3.55068197365047054171469118555, 4.67110312099220426664490383384, 6.15168597703119968137403203861, 7.21216651807741353374990309190, 9.363342330556938292907271739234, 10.12431394420828933306305405309, 10.82465159777444628898558580128, 11.46783059890474748169995753284, 12.15164119525714233589130289284

Graph of the ZZ-function along the critical line