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Properties

Label 2-483-483.11-c1-0-34
Degree 22
Conductor 483483
Sign 0.896+0.442i0.896 + 0.442i
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  + (−0.742 + 1.43i)2-s + (−0.310 + 1.70i)3-s + (−0.361 − 0.507i)4-s + (−1.42 − 1.36i)5-s + (−2.22 − 1.71i)6-s + (−2.18 − 1.49i)7-s + (−2.20 + 0.317i)8-s + (−2.80 − 1.05i)9-s + (3.01 − 1.04i)10-s + (2.80 − 1.44i)11-s + (0.976 − 0.457i)12-s + (0.995 − 1.14i)13-s + (3.77 − 2.02i)14-s + (2.76 − 2.01i)15-s + (1.58 − 4.59i)16-s + (1.57 + 0.150i)17-s + ⋯
L(s)  = 1  + (−0.524 + 1.01i)2-s + (−0.179 + 0.983i)3-s + (−0.180 − 0.253i)4-s + (−0.638 − 0.608i)5-s + (−0.907 − 0.698i)6-s + (−0.824 − 0.565i)7-s + (−0.780 + 0.112i)8-s + (−0.935 − 0.352i)9-s + (0.954 − 0.330i)10-s + (0.846 − 0.436i)11-s + (0.281 − 0.132i)12-s + (0.276 − 0.318i)13-s + (1.00 − 0.542i)14-s + (0.713 − 0.518i)15-s + (0.397 − 1.14i)16-s + (0.382 + 0.0364i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.896+0.442i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.896+0.442i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.896+0.442i0.896 + 0.442i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(11,⋅)\chi_{483} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.896+0.442i)(2,\ 483,\ (\ :1/2),\ 0.896 + 0.442i)

Particular Values

L(1)L(1) ≈\approx 0.346145−0.0807666i0.346145 - 0.0807666i
L(12)L(\frac12) ≈\approx 0.346145−0.0807666i0.346145 - 0.0807666i
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+(0.310−1.70i)T 1 + (0.310 - 1.70i)T
7 1+(2.18+1.49i)T 1 + (2.18 + 1.49i)T
23 1+(3.87+2.82i)T 1 + (3.87 + 2.82i)T
good2 1+(0.742−1.43i)T+(−1.16−1.62i)T2 1 + (0.742 - 1.43i)T + (-1.16 - 1.62i)T^{2}
5 1+(1.42+1.36i)T+(0.237+4.99i)T2 1 + (1.42 + 1.36i)T + (0.237 + 4.99i)T^{2}
11 1+(−2.80+1.44i)T+(6.38−8.96i)T2 1 + (-2.80 + 1.44i)T + (6.38 - 8.96i)T^{2}
13 1+(−0.995+1.14i)T+(−1.85−12.8i)T2 1 + (-0.995 + 1.14i)T + (-1.85 - 12.8i)T^{2}
17 1+(−1.57−0.150i)T+(16.6+3.21i)T2 1 + (-1.57 - 0.150i)T + (16.6 + 3.21i)T^{2}
19 1+(−0.259−2.71i)T+(−18.6+3.59i)T2 1 + (-0.259 - 2.71i)T + (-18.6 + 3.59i)T^{2}
29 1+(1.34−0.613i)T+(18.9−21.9i)T2 1 + (1.34 - 0.613i)T + (18.9 - 21.9i)T^{2}
31 1+(0.952+0.381i)T+(22.4+21.3i)T2 1 + (0.952 + 0.381i)T + (22.4 + 21.3i)T^{2}
37 1+(10.2−2.49i)T+(32.8−16.9i)T2 1 + (10.2 - 2.49i)T + (32.8 - 16.9i)T^{2}
41 1+(3.01+10.2i)T+(−34.4+22.1i)T2 1 + (3.01 + 10.2i)T + (-34.4 + 22.1i)T^{2}
43 1+(2.19+0.315i)T+(41.2+12.1i)T2 1 + (2.19 + 0.315i)T + (41.2 + 12.1i)T^{2}
47 1+(−7.89+4.55i)T+(23.5−40.7i)T2 1 + (-7.89 + 4.55i)T + (23.5 - 40.7i)T^{2}
53 1+(0.972−0.187i)T+(49.2−19.6i)T2 1 + (0.972 - 0.187i)T + (49.2 - 19.6i)T^{2}
59 1+(−9.15+3.16i)T+(46.3−36.4i)T2 1 + (-9.15 + 3.16i)T + (46.3 - 36.4i)T^{2}
61 1+(−1.07−1.36i)T+(−14.3+59.2i)T2 1 + (-1.07 - 1.36i)T + (-14.3 + 59.2i)T^{2}
67 1+(13.8−0.660i)T+(66.6−6.36i)T2 1 + (13.8 - 0.660i)T + (66.6 - 6.36i)T^{2}
71 1+(2.69+4.18i)T+(−29.4+64.5i)T2 1 + (2.69 + 4.18i)T + (-29.4 + 64.5i)T^{2}
73 1+(3.96+5.56i)T+(−23.8+68.9i)T2 1 + (3.96 + 5.56i)T + (-23.8 + 68.9i)T^{2}
79 1+(−1.10+5.72i)T+(−73.3−29.3i)T2 1 + (-1.10 + 5.72i)T + (-73.3 - 29.3i)T^{2}
83 1+(1.52+0.446i)T+(69.8+44.8i)T2 1 + (1.52 + 0.446i)T + (69.8 + 44.8i)T^{2}
89 1+(15.1−6.07i)T+(64.4−61.4i)T2 1 + (15.1 - 6.07i)T + (64.4 - 61.4i)T^{2}
97 1+(−0.967−3.29i)T+(−81.6+52.4i)T2 1 + (-0.967 - 3.29i)T + (-81.6 + 52.4i)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

−10.62022334469812801082633439054, −9.885786870700216175892972431522, −8.833573942322592819037396966129, −8.447155009106057193436596194789, −7.29296077198307629821589209585, −6.29635143666943157927972424510, −5.49244808235760098184636834056, −4.05284053245263992844375242945, −3.37476728848934198379003954650, −0.26327965533270731842727609582, 1.53070524821842892580891950120, 2.75668967268031300770569980035, 3.68766582615290219686643004163, 5.69117841809178262154467945367, 6.59015933156029716109191011768, 7.30495695052684838432460652981, 8.584560058479832713117615133841, 9.341120093605898693328541932192, 10.26084999006071193932086409598, 11.34480647437100202431418757763

Graph of the ZZ-function along the critical line