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Properties

Label 2-968-11.3-c1-0-3
Degree 22
Conductor 968968
Sign −0.995−0.0913i-0.995 - 0.0913i
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  + (0.791 + 2.43i)3-s + (0.454 + 0.330i)5-s + (−1.58 + 4.87i)7-s + (−2.88 + 2.09i)9-s + (−2.52 + 1.83i)13-s + (−0.444 + 1.36i)15-s + (−1.61 − 1.17i)17-s + (−1.23 − 3.80i)19-s − 13.1·21-s + 6.56·23-s + (−1.44 − 4.45i)25-s + (−1.16 − 0.845i)27-s + (0.965 − 2.97i)29-s + (1.16 − 0.845i)31-s + (−2.32 + 1.69i)35-s + ⋯
L(s)  = 1  + (0.457 + 1.40i)3-s + (0.203 + 0.147i)5-s + (−0.598 + 1.84i)7-s + (−0.960 + 0.697i)9-s + (−0.700 + 0.509i)13-s + (−0.114 + 0.353i)15-s + (−0.392 − 0.285i)17-s + (−0.283 − 0.872i)19-s − 2.86·21-s + 1.36·23-s + (−0.289 − 0.891i)25-s + (−0.223 − 0.162i)27-s + (0.179 − 0.551i)29-s + (0.209 − 0.151i)31-s + (−0.393 + 0.285i)35-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=((−0.995−0.0913i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=((−0.995−0.0913i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: −0.995−0.0913i-0.995 - 0.0913i
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: 00
Selberg data: (2, 968, ( :1/2), −0.995−0.0913i)(2,\ 968,\ (\ :1/2),\ -0.995 - 0.0913i)

Particular Values

L(1)L(1) ≈\approx 0.0658170+1.43731i0.0658170 + 1.43731i
L(12)L(\frac12) ≈\approx 0.0658170+1.43731i0.0658170 + 1.43731i
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+(−0.791−2.43i)T+(−2.42+1.76i)T2 1 + (-0.791 - 2.43i)T + (-2.42 + 1.76i)T^{2}
5 1+(−0.454−0.330i)T+(1.54+4.75i)T2 1 + (-0.454 - 0.330i)T + (1.54 + 4.75i)T^{2}
7 1+(1.58−4.87i)T+(−5.66−4.11i)T2 1 + (1.58 - 4.87i)T + (-5.66 - 4.11i)T^{2}
13 1+(2.52−1.83i)T+(4.01−12.3i)T2 1 + (2.52 - 1.83i)T + (4.01 - 12.3i)T^{2}
17 1+(1.61+1.17i)T+(5.25+16.1i)T2 1 + (1.61 + 1.17i)T + (5.25 + 16.1i)T^{2}
19 1+(1.23+3.80i)T+(−15.3+11.1i)T2 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2}
23 1−6.56T+23T2 1 - 6.56T + 23T^{2}
29 1+(−0.965+2.97i)T+(−23.4−17.0i)T2 1 + (-0.965 + 2.97i)T + (-23.4 - 17.0i)T^{2}
31 1+(−1.16+0.845i)T+(9.57−29.4i)T2 1 + (-1.16 + 0.845i)T + (9.57 - 29.4i)T^{2}
37 1+(1.06−3.27i)T+(−29.9−21.7i)T2 1 + (1.06 - 3.27i)T + (-29.9 - 21.7i)T^{2}
41 1+(−2.20−6.77i)T+(−33.1+24.0i)T2 1 + (-2.20 - 6.77i)T + (-33.1 + 24.0i)T^{2}
43 1−1.12T+43T2 1 - 1.12T + 43T^{2}
47 1+(−2.47−7.60i)T+(−38.0+27.6i)T2 1 + (-2.47 - 7.60i)T + (-38.0 + 27.6i)T^{2}
53 1+(−3.43+2.49i)T+(16.3−50.4i)T2 1 + (-3.43 + 2.49i)T + (16.3 - 50.4i)T^{2}
59 1+(3.95−12.1i)T+(−47.7−34.6i)T2 1 + (3.95 - 12.1i)T + (-47.7 - 34.6i)T^{2}
61 1+(−5.76−4.18i)T+(18.8+58.0i)T2 1 + (-5.76 - 4.18i)T + (18.8 + 58.0i)T^{2}
67 1−5.43T+67T2 1 - 5.43T + 67T^{2}
71 1+(2.98+2.16i)T+(21.9+67.5i)T2 1 + (2.98 + 2.16i)T + (21.9 + 67.5i)T^{2}
73 1+(0.965−2.97i)T+(−59.0−42.9i)T2 1 + (0.965 - 2.97i)T + (-59.0 - 42.9i)T^{2}
79 1+(−2.32+1.69i)T+(24.4−75.1i)T2 1 + (-2.32 + 1.69i)T + (24.4 - 75.1i)T^{2}
83 1+(7.38+5.36i)T+(25.6+78.9i)T2 1 + (7.38 + 5.36i)T + (25.6 + 78.9i)T^{2}
89 1+9.68T+89T2 1 + 9.68T + 89T^{2}
97 1+(9.25−6.72i)T+(29.9−92.2i)T2 1 + (9.25 - 6.72i)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

−10.12388986571256275520330525262, −9.460783479990128840940345639677, −9.038423021220904682254738746720, −8.368980515722994205453475904285, −6.90289240095816726126804083918, −5.98834629660488615547599272289, −5.02911929463829169735770419022, −4.34471638160178654180794641615, −2.87618707909981748798648320336, −2.53847686890663629065377261524, 0.61833655461894692779965450623, 1.72180329477055474111526302668, 3.06506352952664786982056685297, 4.03782266709592408977187728519, 5.37501816530791657306081412076, 6.62864613922746453110659849285, 7.11263749826913726339197269878, 7.68100998658943209303194557129, 8.580737090575792104926451815571, 9.630328424169517990415363963144

Graph of the ZZ-function along the critical line