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Properties

Label 2-968-88.53-c1-0-81
Degree 22
Conductor 968968
Sign −0.943+0.330i-0.943 + 0.330i
Analytic cond. 7.729517.72951
Root an. cond. 2.780202.78020
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  + (−0.126 + 1.40i)2-s + (−1.10 − 1.52i)3-s + (−1.96 − 0.357i)4-s + (0.468 − 0.152i)5-s + (2.28 − 1.36i)6-s + (0.141 + 0.102i)7-s + (0.752 − 2.72i)8-s + (−0.165 + 0.508i)9-s + (0.155 + 0.679i)10-s + (1.63 + 3.38i)12-s + (−5.24 − 1.70i)13-s + (−0.162 + 0.186i)14-s + (−0.749 − 0.544i)15-s + (3.74 + 1.40i)16-s + (0.209 + 0.645i)17-s + (−0.694 − 0.296i)18-s + ⋯
L(s)  = 1  + (−0.0896 + 0.995i)2-s + (−0.637 − 0.878i)3-s + (−0.983 − 0.178i)4-s + (0.209 − 0.0681i)5-s + (0.931 − 0.556i)6-s + (0.0534 + 0.0388i)7-s + (0.266 − 0.963i)8-s + (−0.0550 + 0.169i)9-s + (0.0490 + 0.214i)10-s + (0.470 + 0.977i)12-s + (−1.45 − 0.473i)13-s + (−0.0434 + 0.0497i)14-s + (−0.193 − 0.140i)15-s + (0.936 + 0.351i)16-s + (0.0508 + 0.156i)17-s + (−0.163 − 0.0699i)18-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=((−0.943+0.330i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=((−0.943+0.330i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: −0.943+0.330i-0.943 + 0.330i
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ968(493,⋅)\chi_{968} (493, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 968, ( :1/2), −0.943+0.330i)(2,\ 968,\ (\ :1/2),\ -0.943 + 0.330i)

Particular Values

L(1)L(1) ≈\approx 0.0129392−0.0760675i0.0129392 - 0.0760675i
L(12)L(\frac12) ≈\approx 0.0129392−0.0760675i0.0129392 - 0.0760675i
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+(0.126−1.40i)T 1 + (0.126 - 1.40i)T
11 1 1
good3 1+(1.10+1.52i)T+(−0.927+2.85i)T2 1 + (1.10 + 1.52i)T + (-0.927 + 2.85i)T^{2}
5 1+(−0.468+0.152i)T+(4.04−2.93i)T2 1 + (-0.468 + 0.152i)T + (4.04 - 2.93i)T^{2}
7 1+(−0.141−0.102i)T+(2.16+6.65i)T2 1 + (-0.141 - 0.102i)T + (2.16 + 6.65i)T^{2}
13 1+(5.24+1.70i)T+(10.5+7.64i)T2 1 + (5.24 + 1.70i)T + (10.5 + 7.64i)T^{2}
17 1+(−0.209−0.645i)T+(−13.7+9.99i)T2 1 + (-0.209 - 0.645i)T + (-13.7 + 9.99i)T^{2}
19 1+(0.257+0.353i)T+(−5.87+18.0i)T2 1 + (0.257 + 0.353i)T + (-5.87 + 18.0i)T^{2}
23 1−6.88T+23T2 1 - 6.88T + 23T^{2}
29 1+(−0.829+1.14i)T+(−8.96−27.5i)T2 1 + (-0.829 + 1.14i)T + (-8.96 - 27.5i)T^{2}
31 1+(1.29−3.97i)T+(−25.0−18.2i)T2 1 + (1.29 - 3.97i)T + (-25.0 - 18.2i)T^{2}
37 1+(3.83−5.27i)T+(−11.4−35.1i)T2 1 + (3.83 - 5.27i)T + (-11.4 - 35.1i)T^{2}
41 1+(7.91−5.75i)T+(12.6−38.9i)T2 1 + (7.91 - 5.75i)T + (12.6 - 38.9i)T^{2}
43 1+8.22iT−43T2 1 + 8.22iT - 43T^{2}
47 1+(6.27−4.55i)T+(14.5−44.6i)T2 1 + (6.27 - 4.55i)T + (14.5 - 44.6i)T^{2}
53 1+(9.43+3.06i)T+(42.8+31.1i)T2 1 + (9.43 + 3.06i)T + (42.8 + 31.1i)T^{2}
59 1+(−5.55+7.65i)T+(−18.2−56.1i)T2 1 + (-5.55 + 7.65i)T + (-18.2 - 56.1i)T^{2}
61 1+(9.13−2.96i)T+(49.3−35.8i)T2 1 + (9.13 - 2.96i)T + (49.3 - 35.8i)T^{2}
67 1−5.24iT−67T2 1 - 5.24iT - 67T^{2}
71 1+(−1.42−4.39i)T+(−57.4+41.7i)T2 1 + (-1.42 - 4.39i)T + (-57.4 + 41.7i)T^{2}
73 1+(5.54+4.02i)T+(22.5+69.4i)T2 1 + (5.54 + 4.02i)T + (22.5 + 69.4i)T^{2}
79 1+(−2.09+6.44i)T+(−63.9−46.4i)T2 1 + (-2.09 + 6.44i)T + (-63.9 - 46.4i)T^{2}
83 1+(6.92−2.25i)T+(67.1−48.7i)T2 1 + (6.92 - 2.25i)T + (67.1 - 48.7i)T^{2}
89 1+9.61T+89T2 1 + 9.61T + 89T^{2}
97 1+(−1.03+3.19i)T+(−78.4−57.0i)T2 1 + (-1.03 + 3.19i)T + (-78.4 - 57.0i)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

−9.587145000833794110368748790068, −8.615875470144235494064602680163, −7.68633301226799425409840262586, −7.01553956070545587930404276293, −6.42772805997627121034355100907, −5.36112903506028770839389320576, −4.86872239936215907830253333045, −3.29955383869721981986833110388, −1.52352559668601506917384030108, −0.04039506508125898362760160086, 1.89862227439507811441640555890, 3.06493511669006572131544980317, 4.31114715462897188987417629590, 4.88179849375085652770343614094, 5.68144224616576252211765805270, 7.07575199951864399686342103938, 8.058689457945127389240593448526, 9.266717482808656330442634392536, 9.672831063801275937027532375696, 10.44237451408522271684134201883

Graph of the ZZ-function along the critical line