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Properties

Label 2-968-11.3-c1-0-25
Degree 22
Conductor 968968
Sign −0.751−0.659i-0.751 - 0.659i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 1.53i)3-s + (−0.5 − 0.363i)5-s + (−0.5 + 1.53i)7-s + (0.309 − 0.224i)9-s + (−4.73 + 3.44i)13-s + (−0.309 + 0.951i)15-s + (−1.5 − 1.08i)17-s + (−1.5 − 4.61i)19-s + 2.61·21-s − 4·23-s + (−1.42 − 4.39i)25-s + (−4.42 − 3.21i)27-s + (−2.26 + 6.96i)29-s + (−0.881 + 0.640i)31-s + (0.809 − 0.587i)35-s + ⋯
L(s)  = 1  + (−0.288 − 0.888i)3-s + (−0.223 − 0.162i)5-s + (−0.188 + 0.581i)7-s + (0.103 − 0.0748i)9-s + (−1.31 + 0.954i)13-s + (−0.0797 + 0.245i)15-s + (−0.363 − 0.264i)17-s + (−0.344 − 1.05i)19-s + 0.571·21-s − 0.834·23-s + (−0.285 − 0.878i)25-s + (−0.851 − 0.619i)27-s + (−0.420 + 1.29i)29-s + (−0.158 + 0.115i)31-s + (0.136 − 0.0993i)35-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=((−0.751−0.659i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=((−0.751−0.659i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: −0.751−0.659i-0.751 - 0.659i
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ968(729,⋅)\chi_{968} (729, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 968, ( :1/2), −0.751−0.659i)(2,\ 968,\ (\ :1/2),\ -0.751 - 0.659i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
11 1 1
good3 1+(0.5+1.53i)T+(−2.42+1.76i)T2 1 + (0.5 + 1.53i)T + (-2.42 + 1.76i)T^{2}
5 1+(0.5+0.363i)T+(1.54+4.75i)T2 1 + (0.5 + 0.363i)T + (1.54 + 4.75i)T^{2}
7 1+(0.5−1.53i)T+(−5.66−4.11i)T2 1 + (0.5 - 1.53i)T + (-5.66 - 4.11i)T^{2}
13 1+(4.73−3.44i)T+(4.01−12.3i)T2 1 + (4.73 - 3.44i)T + (4.01 - 12.3i)T^{2}
17 1+(1.5+1.08i)T+(5.25+16.1i)T2 1 + (1.5 + 1.08i)T + (5.25 + 16.1i)T^{2}
19 1+(1.5+4.61i)T+(−15.3+11.1i)T2 1 + (1.5 + 4.61i)T + (-15.3 + 11.1i)T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+(2.26−6.96i)T+(−23.4−17.0i)T2 1 + (2.26 - 6.96i)T + (-23.4 - 17.0i)T^{2}
31 1+(0.881−0.640i)T+(9.57−29.4i)T2 1 + (0.881 - 0.640i)T + (9.57 - 29.4i)T^{2}
37 1+(2.97−9.14i)T+(−29.9−21.7i)T2 1 + (2.97 - 9.14i)T + (-29.9 - 21.7i)T^{2}
41 1+(−2.97−9.14i)T+(−33.1+24.0i)T2 1 + (-2.97 - 9.14i)T + (-33.1 + 24.0i)T^{2}
43 1+1.52T+43T2 1 + 1.52T + 43T^{2}
47 1+(3.26+10.0i)T+(−38.0+27.6i)T2 1 + (3.26 + 10.0i)T + (-38.0 + 27.6i)T^{2}
53 1+(0.5−0.363i)T+(16.3−50.4i)T2 1 + (0.5 - 0.363i)T + (16.3 - 50.4i)T^{2}
59 1+(0.736−2.26i)T+(−47.7−34.6i)T2 1 + (0.736 - 2.26i)T + (-47.7 - 34.6i)T^{2}
61 1+(1.5+1.08i)T+(18.8+58.0i)T2 1 + (1.5 + 1.08i)T + (18.8 + 58.0i)T^{2}
67 1+14.4T+67T2 1 + 14.4T + 67T^{2}
71 1+(−4.11−2.99i)T+(21.9+67.5i)T2 1 + (-4.11 - 2.99i)T + (21.9 + 67.5i)T^{2}
73 1+(−0.972+2.99i)T+(−59.0−42.9i)T2 1 + (-0.972 + 2.99i)T + (-59.0 - 42.9i)T^{2}
79 1+(3.11−2.26i)T+(24.4−75.1i)T2 1 + (3.11 - 2.26i)T + (24.4 - 75.1i)T^{2}
83 1+(−7.59−5.51i)T+(25.6+78.9i)T2 1 + (-7.59 - 5.51i)T + (25.6 + 78.9i)T^{2}
89 1+4.47T+89T2 1 + 4.47T + 89T^{2}
97 1+(−9.97+7.24i)T+(29.9−92.2i)T2 1 + (-9.97 + 7.24i)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

−9.436471095503702999469326868188, −8.682853030771989826920917009880, −7.66607476509033615171833985476, −6.86442228364330377677744071102, −6.35484226158552969547950082870, −5.10456946854903446936287782489, −4.28043760848348117393122192835, −2.74198580180955135330202281828, −1.72904588719877146061539163238, 0, 2.12700484838215989704490420027, 3.62351004592595270252987834504, 4.22700366769249206308877111154, 5.27476836842010460240682721543, 6.06458595302149503602470180379, 7.48979605308290102899774872468, 7.70630925125443753922842939066, 9.115879883425577742263852894553, 9.930242040916593437310535614005

Graph of the ZZ-function along the critical line