Properties

Label 2-616-11.4-c1-0-7
Degree 22
Conductor 616616
Sign 0.2810.959i0.281 - 0.959i
Analytic cond. 4.918784.91878
Root an. cond. 2.217832.21783
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  + (−0.645 + 1.98i)3-s + (1.70 − 1.24i)5-s + (−0.309 − 0.951i)7-s + (−1.10 − 0.801i)9-s + (−2.67 + 1.96i)11-s + (4.98 + 3.62i)13-s + (1.36 + 4.19i)15-s + (2.23 − 1.62i)17-s + (−0.142 + 0.438i)19-s + 2.08·21-s + 6.31·23-s + (−0.165 + 0.508i)25-s + (−2.76 + 2.00i)27-s + (0.693 + 2.13i)29-s + (−2.67 − 1.94i)31-s + ⋯
L(s)  = 1  + (−0.372 + 1.14i)3-s + (0.764 − 0.555i)5-s + (−0.116 − 0.359i)7-s + (−0.367 − 0.267i)9-s + (−0.805 + 0.592i)11-s + (1.38 + 1.00i)13-s + (0.352 + 1.08i)15-s + (0.541 − 0.393i)17-s + (−0.0326 + 0.100i)19-s + 0.455·21-s + 1.31·23-s + (−0.0330 + 0.101i)25-s + (−0.532 + 0.386i)27-s + (0.128 + 0.396i)29-s + (−0.480 − 0.349i)31-s + ⋯

Functional equation

Λ(s)=(616s/2ΓC(s)L(s)=((0.2810.959i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(616s/2ΓC(s+1/2)L(s)=((0.2810.959i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 616616    =    237112^{3} \cdot 7 \cdot 11
Sign: 0.2810.959i0.281 - 0.959i
Analytic conductor: 4.918784.91878
Root analytic conductor: 2.217832.21783
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ616(169,)\chi_{616} (169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 616, ( :1/2), 0.2810.959i)(2,\ 616,\ (\ :1/2),\ 0.281 - 0.959i)

Particular Values

L(1)L(1) \approx 1.19364+0.894068i1.19364 + 0.894068i
L(12)L(\frac12) \approx 1.19364+0.894068i1.19364 + 0.894068i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
11 1+(2.671.96i)T 1 + (2.67 - 1.96i)T
good3 1+(0.6451.98i)T+(2.421.76i)T2 1 + (0.645 - 1.98i)T + (-2.42 - 1.76i)T^{2}
5 1+(1.70+1.24i)T+(1.544.75i)T2 1 + (-1.70 + 1.24i)T + (1.54 - 4.75i)T^{2}
13 1+(4.983.62i)T+(4.01+12.3i)T2 1 + (-4.98 - 3.62i)T + (4.01 + 12.3i)T^{2}
17 1+(2.23+1.62i)T+(5.2516.1i)T2 1 + (-2.23 + 1.62i)T + (5.25 - 16.1i)T^{2}
19 1+(0.1420.438i)T+(15.311.1i)T2 1 + (0.142 - 0.438i)T + (-15.3 - 11.1i)T^{2}
23 16.31T+23T2 1 - 6.31T + 23T^{2}
29 1+(0.6932.13i)T+(23.4+17.0i)T2 1 + (-0.693 - 2.13i)T + (-23.4 + 17.0i)T^{2}
31 1+(2.67+1.94i)T+(9.57+29.4i)T2 1 + (2.67 + 1.94i)T + (9.57 + 29.4i)T^{2}
37 1+(2.266.95i)T+(29.9+21.7i)T2 1 + (-2.26 - 6.95i)T + (-29.9 + 21.7i)T^{2}
41 1+(0.0315+0.0970i)T+(33.124.0i)T2 1 + (-0.0315 + 0.0970i)T + (-33.1 - 24.0i)T^{2}
43 13.33T+43T2 1 - 3.33T + 43T^{2}
47 1+(2.567.88i)T+(38.027.6i)T2 1 + (2.56 - 7.88i)T + (-38.0 - 27.6i)T^{2}
53 1+(10.1+7.37i)T+(16.3+50.4i)T2 1 + (10.1 + 7.37i)T + (16.3 + 50.4i)T^{2}
59 1+(0.6361.95i)T+(47.7+34.6i)T2 1 + (-0.636 - 1.95i)T + (-47.7 + 34.6i)T^{2}
61 1+(2.151.56i)T+(18.858.0i)T2 1 + (2.15 - 1.56i)T + (18.8 - 58.0i)T^{2}
67 15.10T+67T2 1 - 5.10T + 67T^{2}
71 1+(11.4+8.31i)T+(21.967.5i)T2 1 + (-11.4 + 8.31i)T + (21.9 - 67.5i)T^{2}
73 1+(0.651+2.00i)T+(59.0+42.9i)T2 1 + (0.651 + 2.00i)T + (-59.0 + 42.9i)T^{2}
79 1+(1.931.40i)T+(24.4+75.1i)T2 1 + (-1.93 - 1.40i)T + (24.4 + 75.1i)T^{2}
83 1+(6.49+4.72i)T+(25.678.9i)T2 1 + (-6.49 + 4.72i)T + (25.6 - 78.9i)T^{2}
89 1+10.9T+89T2 1 + 10.9T + 89T^{2}
97 1+(9.99+7.26i)T+(29.9+92.2i)T2 1 + (9.99 + 7.26i)T + (29.9 + 92.2i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.87409295717971019822567427717, −9.692302646732514764244419132142, −9.505418521297275640777555345690, −8.412982415576980743644359736857, −7.18228643219678665725729532878, −6.05789220271712297991806820819, −5.11924523453322672450319476154, −4.47783737305801165923088126964, −3.29852009936202648505077446282, −1.53160812047034509785150573316, 0.979011188043482530506362601443, 2.37551736154765562121957564007, 3.45602401570214832719401438291, 5.42566359333246566965838394674, 5.97111487880539918218884753873, 6.71064208366805988982927567297, 7.75937103700763838288745068622, 8.495086459075226834662463790238, 9.644783790392192331333720186226, 10.73345042976541322331182293827

Graph of the ZZ-function along the critical line