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Properties

Label 2-968-88.5-c1-0-27
Degree 22
Conductor 968968
Sign −0.662−0.748i-0.662 - 0.748i
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  + (−1.37 − 0.330i)2-s + (−1.38 + 1.90i)3-s + (1.78 + 0.908i)4-s + (3.96 + 1.28i)5-s + (2.53 − 2.16i)6-s + (0.754 − 0.548i)7-s + (−2.14 − 1.83i)8-s + (−0.783 − 2.41i)9-s + (−5.02 − 3.08i)10-s + (−4.19 + 2.13i)12-s + (−2.78 + 0.906i)13-s + (−1.21 + 0.504i)14-s + (−7.93 + 5.76i)15-s + (2.34 + 3.23i)16-s + (−0.755 + 2.32i)17-s + (0.280 + 3.57i)18-s + ⋯
L(s)  = 1  + (−0.972 − 0.233i)2-s + (−0.798 + 1.09i)3-s + (0.890 + 0.454i)4-s + (1.77 + 0.575i)5-s + (1.03 − 0.882i)6-s + (0.285 − 0.207i)7-s + (−0.760 − 0.649i)8-s + (−0.261 − 0.804i)9-s + (−1.58 − 0.974i)10-s + (−1.21 + 0.616i)12-s + (−0.773 + 0.251i)13-s + (−0.325 + 0.134i)14-s + (−2.04 + 1.48i)15-s + (0.587 + 0.809i)16-s + (−0.183 + 0.564i)17-s + (0.0662 + 0.842i)18-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=((−0.662−0.748i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=((−0.662−0.748i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: −0.662−0.748i-0.662 - 0.748i
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ968(269,⋅)\chi_{968} (269, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 968, ( :1/2), −0.662−0.748i)(2,\ 968,\ (\ :1/2),\ -0.662 - 0.748i)

Particular Values

L(1)L(1) ≈\approx 0.366929+0.814693i0.366929 + 0.814693i
L(12)L(\frac12) ≈\approx 0.366929+0.814693i0.366929 + 0.814693i
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+(1.37+0.330i)T 1 + (1.37 + 0.330i)T
11 1 1
good3 1+(1.38−1.90i)T+(−0.927−2.85i)T2 1 + (1.38 - 1.90i)T + (-0.927 - 2.85i)T^{2}
5 1+(−3.96−1.28i)T+(4.04+2.93i)T2 1 + (-3.96 - 1.28i)T + (4.04 + 2.93i)T^{2}
7 1+(−0.754+0.548i)T+(2.16−6.65i)T2 1 + (-0.754 + 0.548i)T + (2.16 - 6.65i)T^{2}
13 1+(2.78−0.906i)T+(10.5−7.64i)T2 1 + (2.78 - 0.906i)T + (10.5 - 7.64i)T^{2}
17 1+(0.755−2.32i)T+(−13.7−9.99i)T2 1 + (0.755 - 2.32i)T + (-13.7 - 9.99i)T^{2}
19 1+(1.57−2.17i)T+(−5.87−18.0i)T2 1 + (1.57 - 2.17i)T + (-5.87 - 18.0i)T^{2}
23 1+3.47T+23T2 1 + 3.47T + 23T^{2}
29 1+(−2.68−3.69i)T+(−8.96+27.5i)T2 1 + (-2.68 - 3.69i)T + (-8.96 + 27.5i)T^{2}
31 1+(1.13+3.47i)T+(−25.0+18.2i)T2 1 + (1.13 + 3.47i)T + (-25.0 + 18.2i)T^{2}
37 1+(2.66+3.67i)T+(−11.4+35.1i)T2 1 + (2.66 + 3.67i)T + (-11.4 + 35.1i)T^{2}
41 1+(−3.33−2.42i)T+(12.6+38.9i)T2 1 + (-3.33 - 2.42i)T + (12.6 + 38.9i)T^{2}
43 1−11.4iT−43T2 1 - 11.4iT - 43T^{2}
47 1+(2.63+1.91i)T+(14.5+44.6i)T2 1 + (2.63 + 1.91i)T + (14.5 + 44.6i)T^{2}
53 1+(−0.618+0.200i)T+(42.8−31.1i)T2 1 + (-0.618 + 0.200i)T + (42.8 - 31.1i)T^{2}
59 1+(−1.71−2.35i)T+(−18.2+56.1i)T2 1 + (-1.71 - 2.35i)T + (-18.2 + 56.1i)T^{2}
61 1+(−10.1−3.30i)T+(49.3+35.8i)T2 1 + (-10.1 - 3.30i)T + (49.3 + 35.8i)T^{2}
67 1−5.42iT−67T2 1 - 5.42iT - 67T^{2}
71 1+(2.39−7.37i)T+(−57.4−41.7i)T2 1 + (2.39 - 7.37i)T + (-57.4 - 41.7i)T^{2}
73 1+(−10.5+7.67i)T+(22.5−69.4i)T2 1 + (-10.5 + 7.67i)T + (22.5 - 69.4i)T^{2}
79 1+(1.49+4.59i)T+(−63.9+46.4i)T2 1 + (1.49 + 4.59i)T + (-63.9 + 46.4i)T^{2}
83 1+(−0.627−0.203i)T+(67.1+48.7i)T2 1 + (-0.627 - 0.203i)T + (67.1 + 48.7i)T^{2}
89 1+2.74T+89T2 1 + 2.74T + 89T^{2}
97 1+(−0.565−1.73i)T+(−78.4+57.0i)T2 1 + (-0.565 - 1.73i)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

−10.23438606963461819028910535901, −9.736100494194554112321466450484, −9.138571474291314892768834607907, −7.942534261942813513032110191726, −6.76259946632224050024683797899, −6.07847044462710051436657670791, −5.31821276708793236830928731463, −4.10338967374603196717284378020, −2.66691184109772253213629045299, −1.65838537086886284476731634679, 0.60587465532577180464264649341, 1.80489541957878857989656709850, 2.42027240981208179538632994679, 5.10353185286490930407267202134, 5.56611188586946606605557288196, 6.49540522480057590015493208788, 6.94825209869219444028688881936, 8.075725912576276023729882743552, 8.911473755775123205084694115086, 9.713529189513693998679251657585

Graph of the ZZ-function along the critical line