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Properties

Label 2-483-23.6-c1-0-4
Degree 22
Conductor 483483
Sign 0.138−0.990i0.138 - 0.990i
Analytic cond. 3.856773.85677
Root an. cond. 1.963861.96386
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0982 − 0.215i)2-s + (−0.959 − 0.281i)3-s + (1.27 + 1.46i)4-s + (−0.841 + 0.540i)5-s + (−0.154 + 0.178i)6-s + (−0.142 + 0.989i)7-s + (0.894 − 0.262i)8-s + (0.841 + 0.540i)9-s + (0.0336 + 0.234i)10-s + (−0.827 − 1.81i)11-s + (−0.807 − 1.76i)12-s + (0.561 + 3.90i)13-s + (0.198 + 0.127i)14-s + (0.959 − 0.281i)15-s + (−0.521 + 3.63i)16-s + (−2.66 + 3.07i)17-s + ⋯
L(s)  = 1  + (0.0694 − 0.152i)2-s + (−0.553 − 0.162i)3-s + (0.636 + 0.734i)4-s + (−0.376 + 0.241i)5-s + (−0.0632 + 0.0729i)6-s + (−0.0537 + 0.374i)7-s + (0.316 − 0.0929i)8-s + (0.280 + 0.180i)9-s + (0.0106 + 0.0740i)10-s + (−0.249 − 0.546i)11-s + (−0.233 − 0.510i)12-s + (0.155 + 1.08i)13-s + (0.0531 + 0.0341i)14-s + (0.247 − 0.0727i)15-s + (−0.130 + 0.907i)16-s + (−0.645 + 0.745i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.138−0.990i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.138−0.990i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.138−0.990i0.138 - 0.990i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(190,⋅)\chi_{483} (190, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.138−0.990i)(2,\ 483,\ (\ :1/2),\ 0.138 - 0.990i)

Particular Values

L(1)L(1) ≈\approx 0.877354+0.763084i0.877354 + 0.763084i
L(12)L(\frac12) ≈\approx 0.877354+0.763084i0.877354 + 0.763084i
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+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
7 1+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
23 1+(−4.79−0.102i)T 1 + (-4.79 - 0.102i)T
good2 1+(−0.0982+0.215i)T+(−1.30−1.51i)T2 1 + (-0.0982 + 0.215i)T + (-1.30 - 1.51i)T^{2}
5 1+(0.841−0.540i)T+(2.07−4.54i)T2 1 + (0.841 - 0.540i)T + (2.07 - 4.54i)T^{2}
11 1+(0.827+1.81i)T+(−7.20+8.31i)T2 1 + (0.827 + 1.81i)T + (-7.20 + 8.31i)T^{2}
13 1+(−0.561−3.90i)T+(−12.4+3.66i)T2 1 + (-0.561 - 3.90i)T + (-12.4 + 3.66i)T^{2}
17 1+(2.66−3.07i)T+(−2.41−16.8i)T2 1 + (2.66 - 3.07i)T + (-2.41 - 16.8i)T^{2}
19 1+(0.0577+0.0666i)T+(−2.70+18.8i)T2 1 + (0.0577 + 0.0666i)T + (-2.70 + 18.8i)T^{2}
29 1+(2.80−3.23i)T+(−4.12−28.7i)T2 1 + (2.80 - 3.23i)T + (-4.12 - 28.7i)T^{2}
31 1+(4.68−1.37i)T+(26.0−16.7i)T2 1 + (4.68 - 1.37i)T + (26.0 - 16.7i)T^{2}
37 1+(−3.51−2.25i)T+(15.3+33.6i)T2 1 + (-3.51 - 2.25i)T + (15.3 + 33.6i)T^{2}
41 1+(−2.35+1.51i)T+(17.0−37.2i)T2 1 + (-2.35 + 1.51i)T + (17.0 - 37.2i)T^{2}
43 1+(−9.65−2.83i)T+(36.1+23.2i)T2 1 + (-9.65 - 2.83i)T + (36.1 + 23.2i)T^{2}
47 1+0.958T+47T2 1 + 0.958T + 47T^{2}
53 1+(−1.66+11.6i)T+(−50.8−14.9i)T2 1 + (-1.66 + 11.6i)T + (-50.8 - 14.9i)T^{2}
59 1+(−0.917−6.38i)T+(−56.6+16.6i)T2 1 + (-0.917 - 6.38i)T + (-56.6 + 16.6i)T^{2}
61 1+(−2.16+0.634i)T+(51.3−32.9i)T2 1 + (-2.16 + 0.634i)T + (51.3 - 32.9i)T^{2}
67 1+(−2.71+5.94i)T+(−43.8−50.6i)T2 1 + (-2.71 + 5.94i)T + (-43.8 - 50.6i)T^{2}
71 1+(−0.811+1.77i)T+(−46.4−53.6i)T2 1 + (-0.811 + 1.77i)T + (-46.4 - 53.6i)T^{2}
73 1+(4.59+5.29i)T+(−10.3+72.2i)T2 1 + (4.59 + 5.29i)T + (-10.3 + 72.2i)T^{2}
79 1+(0.457+3.18i)T+(−75.7+22.2i)T2 1 + (0.457 + 3.18i)T + (-75.7 + 22.2i)T^{2}
83 1+(−4.99−3.21i)T+(34.4+75.4i)T2 1 + (-4.99 - 3.21i)T + (34.4 + 75.4i)T^{2}
89 1+(9.84+2.88i)T+(74.8+48.1i)T2 1 + (9.84 + 2.88i)T + (74.8 + 48.1i)T^{2}
97 1+(−9.57+6.15i)T+(40.2−88.2i)T2 1 + (-9.57 + 6.15i)T + (40.2 - 88.2i)T^{2}
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   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

−11.11202867295133350465941718025, −10.87942883321170900434330431383, −9.312108894345037684724790030498, −8.429224238108240375259218648060, −7.37165836111887146798020916928, −6.69648728355861933159752819487, −5.70037399862928327040327152392, −4.30039497772314103577903157144, −3.25240716672693751519442900865, −1.88413141167935353007406186826, 0.73581791347811696063485459518, 2.52208443657968574104789585375, 4.19353090088370030890266433982, 5.19121068818999484049762517791, 6.01801419724093170100067797946, 7.11416485864219451676374535618, 7.74696483611129106373508782979, 9.191572305582259127953434287463, 10.07657050996634985647280458336, 10.86939715117664898249071385012

Graph of the ZZ-function along the critical line