↑

Properties

Label 2-399-19.4-c1-0-16
Degree 22
Conductor 399399
Sign −0.644+0.764i-0.644 + 0.764i
Analytic cond. 3.186033.18603
Root an. cond. 1.784941.78494
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  + (−0.0505 − 0.0184i)2-s + (−0.173 − 0.984i)3-s + (−1.52 − 1.28i)4-s + (1.59 − 1.33i)5-s + (−0.00934 + 0.0530i)6-s + (0.5 − 0.866i)7-s + (0.107 + 0.186i)8-s + (−0.939 + 0.342i)9-s + (−0.105 + 0.0382i)10-s + (0.760 + 1.31i)11-s + (−0.998 + 1.72i)12-s + (0.789 − 4.48i)13-s + (−0.0412 + 0.0346i)14-s + (−1.59 − 1.33i)15-s + (0.691 + 3.92i)16-s + (−3.16 − 1.15i)17-s + ⋯
L(s)  = 1  + (−0.0357 − 0.0130i)2-s + (−0.100 − 0.568i)3-s + (−0.764 − 0.641i)4-s + (0.712 − 0.597i)5-s + (−0.00381 + 0.0216i)6-s + (0.188 − 0.327i)7-s + (0.0380 + 0.0658i)8-s + (−0.313 + 0.114i)9-s + (−0.0332 + 0.0121i)10-s + (0.229 + 0.397i)11-s + (−0.288 + 0.499i)12-s + (0.219 − 1.24i)13-s + (−0.0110 + 0.00924i)14-s + (−0.411 − 0.345i)15-s + (0.172 + 0.980i)16-s + (−0.767 − 0.279i)17-s + ⋯

Functional equation

Λ(s)=(399s/2 ΓC(s) L(s)=((−0.644+0.764i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(399s/2 ΓC(s+1/2) L(s)=((−0.644+0.764i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 399399    =    3⋅7⋅193 \cdot 7 \cdot 19
Sign: −0.644+0.764i-0.644 + 0.764i
Analytic conductor: 3.186033.18603
Root analytic conductor: 1.784941.78494
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ399(232,⋅)\chi_{399} (232, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 399, ( :1/2), −0.644+0.764i)(2,\ 399,\ (\ :1/2),\ -0.644 + 0.764i)

Particular Values

L(1)L(1) ≈\approx 0.450636−0.968922i0.450636 - 0.968922i
L(12)L(\frac12) ≈\approx 0.450636−0.968922i0.450636 - 0.968922i
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+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
7 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
19 1+(4.14+1.33i)T 1 + (4.14 + 1.33i)T
good2 1+(0.0505+0.0184i)T+(1.53+1.28i)T2 1 + (0.0505 + 0.0184i)T + (1.53 + 1.28i)T^{2}
5 1+(−1.59+1.33i)T+(0.868−4.92i)T2 1 + (-1.59 + 1.33i)T + (0.868 - 4.92i)T^{2}
11 1+(−0.760−1.31i)T+(−5.5+9.52i)T2 1 + (-0.760 - 1.31i)T + (-5.5 + 9.52i)T^{2}
13 1+(−0.789+4.48i)T+(−12.2−4.44i)T2 1 + (-0.789 + 4.48i)T + (-12.2 - 4.44i)T^{2}
17 1+(3.16+1.15i)T+(13.0+10.9i)T2 1 + (3.16 + 1.15i)T + (13.0 + 10.9i)T^{2}
23 1+(4.21+3.53i)T+(3.99+22.6i)T2 1 + (4.21 + 3.53i)T + (3.99 + 22.6i)T^{2}
29 1+(−8.64+3.14i)T+(22.2−18.6i)T2 1 + (-8.64 + 3.14i)T + (22.2 - 18.6i)T^{2}
31 1+(3.00−5.20i)T+(−15.5−26.8i)T2 1 + (3.00 - 5.20i)T + (-15.5 - 26.8i)T^{2}
37 1+6.45T+37T2 1 + 6.45T + 37T^{2}
41 1+(1.00+5.68i)T+(−38.5+14.0i)T2 1 + (1.00 + 5.68i)T + (-38.5 + 14.0i)T^{2}
43 1+(−4.73+3.97i)T+(7.46−42.3i)T2 1 + (-4.73 + 3.97i)T + (7.46 - 42.3i)T^{2}
47 1+(−10.5+3.85i)T+(36.0−30.2i)T2 1 + (-10.5 + 3.85i)T + (36.0 - 30.2i)T^{2}
53 1+(−4.95−4.16i)T+(9.20+52.1i)T2 1 + (-4.95 - 4.16i)T + (9.20 + 52.1i)T^{2}
59 1+(−7.23−2.63i)T+(45.1+37.9i)T2 1 + (-7.23 - 2.63i)T + (45.1 + 37.9i)T^{2}
61 1+(−8.65−7.26i)T+(10.5+60.0i)T2 1 + (-8.65 - 7.26i)T + (10.5 + 60.0i)T^{2}
67 1+(6.76−2.46i)T+(51.3−43.0i)T2 1 + (6.76 - 2.46i)T + (51.3 - 43.0i)T^{2}
71 1+(1.88−1.58i)T+(12.3−69.9i)T2 1 + (1.88 - 1.58i)T + (12.3 - 69.9i)T^{2}
73 1+(1.07+6.11i)T+(−68.5+24.9i)T2 1 + (1.07 + 6.11i)T + (-68.5 + 24.9i)T^{2}
79 1+(−2.83−16.0i)T+(−74.2+27.0i)T2 1 + (-2.83 - 16.0i)T + (-74.2 + 27.0i)T^{2}
83 1+(−7.31+12.6i)T+(−41.5−71.8i)T2 1 + (-7.31 + 12.6i)T + (-41.5 - 71.8i)T^{2}
89 1+(−2.41+13.6i)T+(−83.6−30.4i)T2 1 + (-2.41 + 13.6i)T + (-83.6 - 30.4i)T^{2}
97 1+(−0.905−0.329i)T+(74.3+62.3i)T2 1 + (-0.905 - 0.329i)T + (74.3 + 62.3i)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

−10.54524005234306132283853247078, −10.27585246212294992285990696495, −8.890025647808913464702924446461, −8.525359991109471128031382280721, −7.11133240783600103495294142556, −6.01114802445430223906806364898, −5.20792206397051042149104028885, −4.18336266685608268221872343970, −2.15732995376620869783055649363, −0.74304169467410138209579392953, 2.27302161535032249318531494148, 3.74108657541421354623076555791, 4.57872155581394607366559782514, 5.89744209736651973888604593775, 6.75598781826224586502926066509, 8.214857990529205273562280455414, 8.931901443570002734973452283766, 9.691811926310234975277084371782, 10.62248602633282055274013388814, 11.55970013236213263138202148355

Graph of the ZZ-function along the critical line