↑

Properties

Label 2-55-55.14-c1-0-2
Degree 22
Conductor 5555
Sign 0.245+0.969i0.245 + 0.969i
Analytic cond. 0.4391770.439177
Root an. cond. 0.6627040.662704
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  + (−0.725 − 0.998i)2-s + (−0.346 + 0.112i)3-s + (0.147 − 0.453i)4-s + (0.0238 − 2.23i)5-s + (0.363 + 0.264i)6-s + (2.45 + 0.798i)7-s + (−2.90 + 0.944i)8-s + (−2.31 + 1.68i)9-s + (−2.24 + 1.59i)10-s + (3.12 + 1.12i)11-s + 0.173i·12-s + (1.62 + 2.23i)13-s + (−0.985 − 3.03i)14-s + (0.243 + 0.776i)15-s + (2.27 + 1.65i)16-s + (−2.26 + 3.11i)17-s + ⋯
L(s)  = 1  + (−0.512 − 0.705i)2-s + (−0.199 + 0.0649i)3-s + (0.0737 − 0.226i)4-s + (0.0106 − 0.999i)5-s + (0.148 + 0.107i)6-s + (0.929 + 0.301i)7-s + (−1.02 + 0.333i)8-s + (−0.773 + 0.561i)9-s + (−0.711 + 0.505i)10-s + (0.940 + 0.339i)11-s + 0.0501i·12-s + (0.449 + 0.619i)13-s + (−0.263 − 0.810i)14-s + (0.0628 + 0.200i)15-s + (0.569 + 0.414i)16-s + (−0.549 + 0.755i)17-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=((0.245+0.969i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+1/2) L(s)=((0.245+0.969i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 0.245+0.969i0.245 + 0.969i
Analytic conductor: 0.4391770.439177
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ55(14,⋅)\chi_{55} (14, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1/2), 0.245+0.969i)(2,\ 55,\ (\ :1/2),\ 0.245 + 0.969i)

Particular Values

L(1)L(1) ≈\approx 0.537278−0.418191i0.537278 - 0.418191i
L(12)L(\frac12) ≈\approx 0.537278−0.418191i0.537278 - 0.418191i
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)
bad5 1+(−0.0238+2.23i)T 1 + (-0.0238 + 2.23i)T
11 1+(−3.12−1.12i)T 1 + (-3.12 - 1.12i)T
good2 1+(0.725+0.998i)T+(−0.618+1.90i)T2 1 + (0.725 + 0.998i)T + (-0.618 + 1.90i)T^{2}
3 1+(0.346−0.112i)T+(2.42−1.76i)T2 1 + (0.346 - 0.112i)T + (2.42 - 1.76i)T^{2}
7 1+(−2.45−0.798i)T+(5.66+4.11i)T2 1 + (-2.45 - 0.798i)T + (5.66 + 4.11i)T^{2}
13 1+(−1.62−2.23i)T+(−4.01+12.3i)T2 1 + (-1.62 - 2.23i)T + (-4.01 + 12.3i)T^{2}
17 1+(2.26−3.11i)T+(−5.25−16.1i)T2 1 + (2.26 - 3.11i)T + (-5.25 - 16.1i)T^{2}
19 1+(−0.0857−0.264i)T+(−15.3+11.1i)T2 1 + (-0.0857 - 0.264i)T + (-15.3 + 11.1i)T^{2}
23 1+8.40iT−23T2 1 + 8.40iT - 23T^{2}
29 1+(1.02−3.16i)T+(−23.4−17.0i)T2 1 + (1.02 - 3.16i)T + (-23.4 - 17.0i)T^{2}
31 1+(0.456−0.331i)T+(9.57−29.4i)T2 1 + (0.456 - 0.331i)T + (9.57 - 29.4i)T^{2}
37 1+(−0.497−0.161i)T+(29.9+21.7i)T2 1 + (-0.497 - 0.161i)T + (29.9 + 21.7i)T^{2}
41 1+(1.57+4.86i)T+(−33.1+24.0i)T2 1 + (1.57 + 4.86i)T + (-33.1 + 24.0i)T^{2}
43 1−2.54iT−43T2 1 - 2.54iT - 43T^{2}
47 1+(4.68−1.52i)T+(38.0−27.6i)T2 1 + (4.68 - 1.52i)T + (38.0 - 27.6i)T^{2}
53 1+(−5.12−7.05i)T+(−16.3+50.4i)T2 1 + (-5.12 - 7.05i)T + (-16.3 + 50.4i)T^{2}
59 1+(2.31−7.13i)T+(−47.7−34.6i)T2 1 + (2.31 - 7.13i)T + (-47.7 - 34.6i)T^{2}
61 1+(11.4+8.33i)T+(18.8+58.0i)T2 1 + (11.4 + 8.33i)T + (18.8 + 58.0i)T^{2}
67 1+3.20iT−67T2 1 + 3.20iT - 67T^{2}
71 1+(−6.79−4.93i)T+(21.9+67.5i)T2 1 + (-6.79 - 4.93i)T + (21.9 + 67.5i)T^{2}
73 1+(12.3+4.02i)T+(59.0+42.9i)T2 1 + (12.3 + 4.02i)T + (59.0 + 42.9i)T^{2}
79 1+(−7.85+5.70i)T+(24.4−75.1i)T2 1 + (-7.85 + 5.70i)T + (24.4 - 75.1i)T^{2}
83 1+(−1.93+2.66i)T+(−25.6−78.9i)T2 1 + (-1.93 + 2.66i)T + (-25.6 - 78.9i)T^{2}
89 1+2.48T+89T2 1 + 2.48T + 89T^{2}
97 1+(6.40+8.81i)T+(−29.9+92.2i)T2 1 + (6.40 + 8.81i)T + (-29.9 + 92.2i)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

−14.99985509861852945895134498871, −14.06990911915570951937399001024, −12.40289482625520856623690217809, −11.53657428751053321174025869808, −10.64070313557157476748871117909, −9.054205466256559674275725971206, −8.449211174199302255754010612488, −6.10874749854609150362245226240, −4.66639998614416985705782401760, −1.80567855218291239778268290432, 3.40210840687127325413544219627, 5.92194244313899952426756250319, 7.05469794989621774496536166490, 8.178708233557622709371050113487, 9.437744752438169856331928753427, 11.24142654725924826127867066251, 11.71964908879812819489883028120, 13.64579366618590351485027552149, 14.72243232705206917055039392514, 15.51318585884385469720715893123

Graph of the ZZ-function along the critical line