↑

Properties

Label 2-42e2-9.4-c1-0-16
Degree 22
Conductor 17641764
Sign 0.904−0.426i0.904 - 0.426i
Analytic cond. 14.085614.0856
Root an. cond. 3.753083.75308
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  + (1.05 − 1.37i)3-s + (−0.736 + 1.27i)5-s + (−0.788 − 2.89i)9-s + (2.35 + 4.07i)11-s + (−1.23 + 2.13i)13-s + (0.981 + 2.35i)15-s + 2.88·17-s + 0.763·19-s + (1.12 − 1.94i)23-s + (1.41 + 2.44i)25-s + (−4.81 − 1.95i)27-s + (−0.583 − 1.01i)29-s + (−4.06 + 7.04i)31-s + (8.08 + 1.04i)33-s − 1.62·37-s + ⋯
L(s)  = 1  + (0.607 − 0.794i)3-s + (−0.329 + 0.570i)5-s + (−0.262 − 0.964i)9-s + (0.709 + 1.22i)11-s + (−0.341 + 0.592i)13-s + (0.253 + 0.608i)15-s + 0.698·17-s + 0.175·19-s + (0.233 − 0.404i)23-s + (0.282 + 0.489i)25-s + (−0.926 − 0.376i)27-s + (−0.108 − 0.187i)29-s + (−0.730 + 1.26i)31-s + (1.40 + 0.182i)33-s − 0.266·37-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((0.904−0.426i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s+1/2) L(s)=((0.904−0.426i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 0.904−0.426i0.904 - 0.426i
Analytic conductor: 14.085614.0856
Root analytic conductor: 3.753083.75308
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1764(589,⋅)\chi_{1764} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :1/2), 0.904−0.426i)(2,\ 1764,\ (\ :1/2),\ 0.904 - 0.426i)

Particular Values

L(1)L(1) ≈\approx 1.9728001001.972800100
L(12)L(\frac12) ≈\approx 1.9728001001.972800100
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 1
3 1+(−1.05+1.37i)T 1 + (-1.05 + 1.37i)T
7 1 1
good5 1+(0.736−1.27i)T+(−2.5−4.33i)T2 1 + (0.736 - 1.27i)T + (-2.5 - 4.33i)T^{2}
11 1+(−2.35−4.07i)T+(−5.5+9.52i)T2 1 + (-2.35 - 4.07i)T + (-5.5 + 9.52i)T^{2}
13 1+(1.23−2.13i)T+(−6.5−11.2i)T2 1 + (1.23 - 2.13i)T + (-6.5 - 11.2i)T^{2}
17 1−2.88T+17T2 1 - 2.88T + 17T^{2}
19 1−0.763T+19T2 1 - 0.763T + 19T^{2}
23 1+(−1.12+1.94i)T+(−11.5−19.9i)T2 1 + (-1.12 + 1.94i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.583+1.01i)T+(−14.5+25.1i)T2 1 + (0.583 + 1.01i)T + (-14.5 + 25.1i)T^{2}
31 1+(4.06−7.04i)T+(−15.5−26.8i)T2 1 + (4.06 - 7.04i)T + (-15.5 - 26.8i)T^{2}
37 1+1.62T+37T2 1 + 1.62T + 37T^{2}
41 1+(1.37−2.38i)T+(−20.5−35.5i)T2 1 + (1.37 - 2.38i)T + (-20.5 - 35.5i)T^{2}
43 1+(−4.93−8.55i)T+(−21.5+37.2i)T2 1 + (-4.93 - 8.55i)T + (-21.5 + 37.2i)T^{2}
47 1+(0.406+0.704i)T+(−23.5+40.7i)T2 1 + (0.406 + 0.704i)T + (-23.5 + 40.7i)T^{2}
53 1−13.1T+53T2 1 - 13.1T + 53T^{2}
59 1+(0.593−1.02i)T+(−29.5−51.0i)T2 1 + (0.593 - 1.02i)T + (-29.5 - 51.0i)T^{2}
61 1+(−4.30−7.45i)T+(−30.5+52.8i)T2 1 + (-4.30 - 7.45i)T + (-30.5 + 52.8i)T^{2}
67 1+(−6.00+10.4i)T+(−33.5−58.0i)T2 1 + (-6.00 + 10.4i)T + (-33.5 - 58.0i)T^{2}
71 1−9.31T+71T2 1 - 9.31T + 71T^{2}
73 1−16.1T+73T2 1 - 16.1T + 73T^{2}
79 1+(−6.14−10.6i)T+(−39.5+68.4i)T2 1 + (-6.14 - 10.6i)T + (-39.5 + 68.4i)T^{2}
83 1+(8.36+14.4i)T+(−41.5+71.8i)T2 1 + (8.36 + 14.4i)T + (-41.5 + 71.8i)T^{2}
89 1+14.3T+89T2 1 + 14.3T + 89T^{2}
97 1+(3.35+5.81i)T+(−48.5+84.0i)T2 1 + (3.35 + 5.81i)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

−9.342540279255162568009468230804, −8.518250601936546941235918447061, −7.58429920365149922705541002164, −7.02151384634562822009846919511, −6.57070785057751701848767918644, −5.31751771978489603661101618978, −4.18043644921588831470435546634, −3.32493700342369386432188322078, −2.31885024556162471864056995112, −1.29653954849416531084490055388, 0.76225284782453407957670161942, 2.39478651645325707085980130162, 3.54073004453757510665710034666, 3.98888268478179936792139278375, 5.23335757918541685946464639957, 5.67340719256537401192259357353, 7.00716100143022713037310855276, 7.979596646657312015826589567949, 8.480733997686995913424423463686, 9.203223458450899461328836411534

Graph of the ZZ-function along the critical line