↑

Properties

Label 2-45-45.38-c1-0-2
Degree 22
Conductor 4545
Sign 0.985+0.170i0.985 + 0.170i
Analytic cond. 0.3593260.359326
Root an. cond. 0.5994380.599438
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  + (0.186 − 0.0499i)2-s + (1.53 − 0.806i)3-s + (−1.69 + 0.981i)4-s + (−2.04 − 0.893i)5-s + (0.245 − 0.226i)6-s + (0.632 + 2.35i)7-s + (−0.540 + 0.540i)8-s + (1.69 − 2.47i)9-s + (−0.426 − 0.0641i)10-s + (−2.14 − 1.23i)11-s + (−1.81 + 2.87i)12-s + (−0.422 + 1.57i)13-s + (0.235 + 0.407i)14-s + (−3.86 + 0.282i)15-s + (1.88 − 3.27i)16-s + (0.403 + 0.403i)17-s + ⋯
L(s)  = 1  + (0.131 − 0.0352i)2-s + (0.885 − 0.465i)3-s + (−0.849 + 0.490i)4-s + (−0.916 − 0.399i)5-s + (0.100 − 0.0925i)6-s + (0.238 + 0.891i)7-s + (−0.191 + 0.191i)8-s + (0.566 − 0.823i)9-s + (−0.134 − 0.0202i)10-s + (−0.646 − 0.373i)11-s + (−0.523 + 0.829i)12-s + (−0.117 + 0.436i)13-s + (0.0629 + 0.108i)14-s + (−0.997 + 0.0728i)15-s + (0.472 − 0.818i)16-s + (0.0979 + 0.0979i)17-s + ⋯

Functional equation

Λ(s)=(45s/2 ΓC(s) L(s)=((0.985+0.170i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(45s/2 ΓC(s+1/2) L(s)=((0.985+0.170i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4545    =    32⋅53^{2} \cdot 5
Sign: 0.985+0.170i0.985 + 0.170i
Analytic conductor: 0.3593260.359326
Root analytic conductor: 0.5994380.599438
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ45(38,⋅)\chi_{45} (38, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 45, ( :1/2), 0.985+0.170i)(2,\ 45,\ (\ :1/2),\ 0.985 + 0.170i)

Particular Values

L(1)L(1) ≈\approx 0.843395−0.0722655i0.843395 - 0.0722655i
L(12)L(\frac12) ≈\approx 0.843395−0.0722655i0.843395 - 0.0722655i
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+(−1.53+0.806i)T 1 + (-1.53 + 0.806i)T
5 1+(2.04+0.893i)T 1 + (2.04 + 0.893i)T
good2 1+(−0.186+0.0499i)T+(1.73−i)T2 1 + (-0.186 + 0.0499i)T + (1.73 - i)T^{2}
7 1+(−0.632−2.35i)T+(−6.06+3.5i)T2 1 + (-0.632 - 2.35i)T + (-6.06 + 3.5i)T^{2}
11 1+(2.14+1.23i)T+(5.5+9.52i)T2 1 + (2.14 + 1.23i)T + (5.5 + 9.52i)T^{2}
13 1+(0.422−1.57i)T+(−11.2−6.5i)T2 1 + (0.422 - 1.57i)T + (-11.2 - 6.5i)T^{2}
17 1+(−0.403−0.403i)T+17iT2 1 + (-0.403 - 0.403i)T + 17iT^{2}
19 1+4.28iT−19T2 1 + 4.28iT - 19T^{2}
23 1+(−6.82−1.82i)T+(19.9+11.5i)T2 1 + (-6.82 - 1.82i)T + (19.9 + 11.5i)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+(1.97+3.41i)T+(−15.5+26.8i)T2 1 + (1.97 + 3.41i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.171−0.171i)T−37iT2 1 + (0.171 - 0.171i)T - 37iT^{2}
41 1+(6.52−3.76i)T+(20.5−35.5i)T2 1 + (6.52 - 3.76i)T + (20.5 - 35.5i)T^{2}
43 1+(4.95−1.32i)T+(37.2−21.5i)T2 1 + (4.95 - 1.32i)T + (37.2 - 21.5i)T^{2}
47 1+(−2.91+0.780i)T+(40.7−23.5i)T2 1 + (-2.91 + 0.780i)T + (40.7 - 23.5i)T^{2}
53 1+(6.12−6.12i)T−53iT2 1 + (6.12 - 6.12i)T - 53iT^{2}
59 1+(2.27+3.93i)T+(−29.5+51.0i)T2 1 + (2.27 + 3.93i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.235−0.408i)T+(−30.5−52.8i)T2 1 + (0.235 - 0.408i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.65−0.443i)T+(58.0+33.5i)T2 1 + (-1.65 - 0.443i)T + (58.0 + 33.5i)T^{2}
71 1+3.50iT−71T2 1 + 3.50iT - 71T^{2}
73 1+(−6.88−6.88i)T+73iT2 1 + (-6.88 - 6.88i)T + 73iT^{2}
79 1+(6.50+3.75i)T+(39.5+68.4i)T2 1 + (6.50 + 3.75i)T + (39.5 + 68.4i)T^{2}
83 1+(2.85+10.6i)T+(−71.8+41.5i)T2 1 + (2.85 + 10.6i)T + (-71.8 + 41.5i)T^{2}
89 1−2.90T+89T2 1 - 2.90T + 89T^{2}
97 1+(0.379+1.41i)T+(−84.0+48.5i)T2 1 + (0.379 + 1.41i)T + (-84.0 + 48.5i)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

−15.57054408431111275645791441484, −14.71341217093815769545143736435, −13.34866655699360315306199241637, −12.63769672860124598186814695771, −11.49596861863144305441438697106, −9.179237388731723501579144344548, −8.544340175786030015721421855936, −7.40425802778464981417182416787, −4.92754752185020384520696212780, −3.21939218442922471925428814424, 3.61896249127040887007475587932, 4.86688862183632899891884717176, 7.39477596190944703343819281460, 8.456429090018229660350434918857, 9.976005614845232100625842974184, 10.78940850557615796307973086992, 12.75974420927033405520912497223, 13.85471825835248330716911031787, 14.78597163194048070104720306109, 15.48759207724960288791790452513

Graph of the ZZ-function along the critical line