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Properties

Label 2-968-88.51-c1-0-31
Degree 22
Conductor 968968
Sign −0.860−0.508i-0.860 - 0.508i
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  + (0.468 + 1.33i)2-s + (0.127 − 0.393i)3-s + (−1.56 + 1.25i)4-s + (1.55 + 2.14i)5-s + (0.585 − 0.0138i)6-s + (1.01 + 3.12i)7-s + (−2.40 − 1.49i)8-s + (2.28 + 1.66i)9-s + (−2.12 + 3.07i)10-s + (0.292 + 0.774i)12-s + (1.10 + 0.801i)13-s + (−3.69 + 2.82i)14-s + (1.04 − 0.338i)15-s + (0.871 − 3.90i)16-s + (−3.22 − 4.44i)17-s + (−1.14 + 3.83i)18-s + ⋯
L(s)  = 1  + (0.331 + 0.943i)2-s + (0.0739 − 0.227i)3-s + (−0.780 + 0.625i)4-s + (0.695 + 0.957i)5-s + (0.239 − 0.00564i)6-s + (0.384 + 1.18i)7-s + (−0.848 − 0.529i)8-s + (0.762 + 0.554i)9-s + (−0.672 + 0.973i)10-s + (0.0845 + 0.223i)12-s + (0.305 + 0.222i)13-s + (−0.988 + 0.754i)14-s + (0.269 − 0.0874i)15-s + (0.217 − 0.975i)16-s + (−0.782 − 1.07i)17-s + (−0.270 + 0.903i)18-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=((−0.860−0.508i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=((−0.860−0.508i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: −0.860−0.508i-0.860 - 0.508i
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ968(403,⋅)\chi_{968} (403, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 968, ( :1/2), −0.860−0.508i)(2,\ 968,\ (\ :1/2),\ -0.860 - 0.508i)

Particular Values

L(1)L(1) ≈\approx 0.549030+2.00825i0.549030 + 2.00825i
L(12)L(\frac12) ≈\approx 0.549030+2.00825i0.549030 + 2.00825i
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+(−0.468−1.33i)T 1 + (-0.468 - 1.33i)T
11 1 1
good3 1+(−0.127+0.393i)T+(−2.42−1.76i)T2 1 + (-0.127 + 0.393i)T + (-2.42 - 1.76i)T^{2}
5 1+(−1.55−2.14i)T+(−1.54+4.75i)T2 1 + (-1.55 - 2.14i)T + (-1.54 + 4.75i)T^{2}
7 1+(−1.01−3.12i)T+(−5.66+4.11i)T2 1 + (-1.01 - 3.12i)T + (-5.66 + 4.11i)T^{2}
13 1+(−1.10−0.801i)T+(4.01+12.3i)T2 1 + (-1.10 - 0.801i)T + (4.01 + 12.3i)T^{2}
17 1+(3.22+4.44i)T+(−5.25+16.1i)T2 1 + (3.22 + 4.44i)T + (-5.25 + 16.1i)T^{2}
19 1+(−2.16−0.702i)T+(15.3+11.1i)T2 1 + (-2.16 - 0.702i)T + (15.3 + 11.1i)T^{2}
23 1−6.38iT−23T2 1 - 6.38iT - 23T^{2}
29 1+(2.03+6.25i)T+(−23.4+17.0i)T2 1 + (2.03 + 6.25i)T + (-23.4 + 17.0i)T^{2}
31 1+(−0.644+0.886i)T+(−9.57−29.4i)T2 1 + (-0.644 + 0.886i)T + (-9.57 - 29.4i)T^{2}
37 1+(−4.60+1.49i)T+(29.9−21.7i)T2 1 + (-4.60 + 1.49i)T + (29.9 - 21.7i)T^{2}
41 1+(9.54+3.10i)T+(33.1+24.0i)T2 1 + (9.54 + 3.10i)T + (33.1 + 24.0i)T^{2}
43 1−6.43iT−43T2 1 - 6.43iT - 43T^{2}
47 1+(38.0+27.6i)T2 1 + (38.0 + 27.6i)T^{2}
53 1+(−4.39+6.05i)T+(−16.3−50.4i)T2 1 + (-4.39 + 6.05i)T + (-16.3 - 50.4i)T^{2}
59 1+(−1.00−3.08i)T+(−47.7+34.6i)T2 1 + (-1.00 - 3.08i)T + (-47.7 + 34.6i)T^{2}
61 1+(5.32−3.86i)T+(18.8−58.0i)T2 1 + (5.32 - 3.86i)T + (18.8 - 58.0i)T^{2}
67 1+6.07T+67T2 1 + 6.07T + 67T^{2}
71 1+(−3.75−5.16i)T+(−21.9+67.5i)T2 1 + (-3.75 - 5.16i)T + (-21.9 + 67.5i)T^{2}
73 1+(0.895−0.291i)T+(59.0−42.9i)T2 1 + (0.895 - 0.291i)T + (59.0 - 42.9i)T^{2}
79 1+(7.52+5.47i)T+(24.4+75.1i)T2 1 + (7.52 + 5.47i)T + (24.4 + 75.1i)T^{2}
83 1+(2.67+3.67i)T+(−25.6+78.9i)T2 1 + (2.67 + 3.67i)T + (-25.6 + 78.9i)T^{2}
89 1+10.6T+89T2 1 + 10.6T + 89T^{2}
97 1+(6.05+4.39i)T+(29.9+92.2i)T2 1 + (6.05 + 4.39i)T + (29.9 + 92.2i)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

−10.06043375835499979002665864041, −9.451817193616802518202809020990, −8.579134900816126194042827497066, −7.60402579831793708834126097098, −6.98188034818580229595493203460, −6.07237512723853996052702417139, −5.40154584210946109070757557201, −4.41222160833041869775310407468, −3.01269589983836362839155111145, −2.02607594377501071671567066404, 0.935901505132151969578071689686, 1.76866704490821651382637371663, 3.41644320479774307748960358099, 4.34588218820055001092987294609, 4.87134089549927908463931542257, 6.02256470789207770226432167345, 7.03078242759784812872314537135, 8.421202244920402366460799809331, 8.977156602540985906685611222784, 9.904723093603391400765959769864

Graph of the ZZ-function along the critical line