↑

Properties

Label 2-92-23.16-c5-0-7
Degree 22
Conductor 9292
Sign −0.142+0.989i-0.142 + 0.989i
Analytic cond. 14.755314.7553
Root an. cond. 3.841263.84126
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−7.14 + 4.59i)3-s + (−4.30 − 9.42i)5-s + (54.2 − 15.9i)7-s + (−70.9 + 155. i)9-s + (−60.7 − 70.1i)11-s + (−336. − 98.9i)13-s + (73.9 + 47.5i)15-s + (297. − 2.06e3i)17-s + (−62.3 − 433. i)19-s + (−314. + 363. i)21-s + (−30.3 − 2.53e3i)23-s + (1.97e3 − 2.28e3i)25-s + (−500. − 3.47e3i)27-s + (−294. + 2.04e3i)29-s + (−6.24e3 − 4.01e3i)31-s + ⋯
L(s)  = 1  + (−0.458 + 0.294i)3-s + (−0.0769 − 0.168i)5-s + (0.418 − 0.122i)7-s + (−0.292 + 0.639i)9-s + (−0.151 − 0.174i)11-s + (−0.552 − 0.162i)13-s + (0.0848 + 0.0545i)15-s + (0.249 − 1.73i)17-s + (−0.0396 − 0.275i)19-s + (−0.155 + 0.179i)21-s + (−0.0119 − 0.999i)23-s + (0.632 − 0.729i)25-s + (−0.132 − 0.918i)27-s + (−0.0649 + 0.451i)29-s + (−1.16 − 0.750i)31-s + ⋯

Functional equation

Λ(s)=(92s/2 ΓC(s) L(s)=((−0.142+0.989i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(92s/2 ΓC(s+5/2) L(s)=((−0.142+0.989i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9292    =    22⋅232^{2} \cdot 23
Sign: −0.142+0.989i-0.142 + 0.989i
Analytic conductor: 14.755314.7553
Root analytic conductor: 3.841263.84126
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ92(85,⋅)\chi_{92} (85, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 92, ( :5/2), −0.142+0.989i)(2,\ 92,\ (\ :5/2),\ -0.142 + 0.989i)

Particular Values

L(3)L(3) ≈\approx 0.588581−0.679131i0.588581 - 0.679131i
L(12)L(\frac12) ≈\approx 0.588581−0.679131i0.588581 - 0.679131i
L(72)L(\frac{7}{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
23 1+(30.3+2.53e3i)T 1 + (30.3 + 2.53e3i)T
good3 1+(7.14−4.59i)T+(100.−221.i)T2 1 + (7.14 - 4.59i)T + (100. - 221. i)T^{2}
5 1+(4.30+9.42i)T+(−2.04e3+2.36e3i)T2 1 + (4.30 + 9.42i)T + (-2.04e3 + 2.36e3i)T^{2}
7 1+(−54.2+15.9i)T+(1.41e4−9.08e3i)T2 1 + (-54.2 + 15.9i)T + (1.41e4 - 9.08e3i)T^{2}
11 1+(60.7+70.1i)T+(−2.29e4+1.59e5i)T2 1 + (60.7 + 70.1i)T + (-2.29e4 + 1.59e5i)T^{2}
13 1+(336.+98.9i)T+(3.12e5+2.00e5i)T2 1 + (336. + 98.9i)T + (3.12e5 + 2.00e5i)T^{2}
17 1+(−297.+2.06e3i)T+(−1.36e6−4.00e5i)T2 1 + (-297. + 2.06e3i)T + (-1.36e6 - 4.00e5i)T^{2}
19 1+(62.3+433.i)T+(−2.37e6+6.97e5i)T2 1 + (62.3 + 433. i)T + (-2.37e6 + 6.97e5i)T^{2}
29 1+(294.−2.04e3i)T+(−1.96e7−5.77e6i)T2 1 + (294. - 2.04e3i)T + (-1.96e7 - 5.77e6i)T^{2}
31 1+(6.24e3+4.01e3i)T+(1.18e7+2.60e7i)T2 1 + (6.24e3 + 4.01e3i)T + (1.18e7 + 2.60e7i)T^{2}
37 1+(−4.19e3+9.17e3i)T+(−4.54e7−5.24e7i)T2 1 + (-4.19e3 + 9.17e3i)T + (-4.54e7 - 5.24e7i)T^{2}
41 1+(−316.−694.i)T+(−7.58e7+8.75e7i)T2 1 + (-316. - 694. i)T + (-7.58e7 + 8.75e7i)T^{2}
43 1+(7.74e3−4.97e3i)T+(6.10e7−1.33e8i)T2 1 + (7.74e3 - 4.97e3i)T + (6.10e7 - 1.33e8i)T^{2}
47 1+9.13e3T+2.29e8T2 1 + 9.13e3T + 2.29e8T^{2}
53 1+(−2.85e4+8.38e3i)T+(3.51e8−2.26e8i)T2 1 + (-2.85e4 + 8.38e3i)T + (3.51e8 - 2.26e8i)T^{2}
59 1+(7.86e3+2.30e3i)T+(6.01e8+3.86e8i)T2 1 + (7.86e3 + 2.30e3i)T + (6.01e8 + 3.86e8i)T^{2}
61 1+(−5.63e3−3.62e3i)T+(3.50e8+7.68e8i)T2 1 + (-5.63e3 - 3.62e3i)T + (3.50e8 + 7.68e8i)T^{2}
67 1+(−930.+1.07e3i)T+(−1.92e8−1.33e9i)T2 1 + (-930. + 1.07e3i)T + (-1.92e8 - 1.33e9i)T^{2}
71 1+(1.44e4−1.66e4i)T+(−2.56e8−1.78e9i)T2 1 + (1.44e4 - 1.66e4i)T + (-2.56e8 - 1.78e9i)T^{2}
73 1+(−6.81e3−4.74e4i)T+(−1.98e9+5.84e8i)T2 1 + (-6.81e3 - 4.74e4i)T + (-1.98e9 + 5.84e8i)T^{2}
79 1+(−2.75e4−8.09e3i)T+(2.58e9+1.66e9i)T2 1 + (-2.75e4 - 8.09e3i)T + (2.58e9 + 1.66e9i)T^{2}
83 1+(3.80e4−8.33e4i)T+(−2.57e9−2.97e9i)T2 1 + (3.80e4 - 8.33e4i)T + (-2.57e9 - 2.97e9i)T^{2}
89 1+(−6.39e4+4.11e4i)T+(2.31e9−5.07e9i)T2 1 + (-6.39e4 + 4.11e4i)T + (2.31e9 - 5.07e9i)T^{2}
97 1+(4.13e3+9.05e3i)T+(−5.62e9+6.48e9i)T2 1 + (4.13e3 + 9.05e3i)T + (-5.62e9 + 6.48e9i)T^{2}
show more
show less
   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

−12.73927390238194171342992077400, −11.56938416604228011902257038797, −10.79658813816145235325298166872, −9.612766550735735208353370441627, −8.286510721934009854975625046423, −7.12406485815913856946958122937, −5.46042895707427541036345511284, −4.58117632003368084224203055969, −2.56963910741586884545490264652, −0.38642455630537674345015378047, 1.56520459256704456833175817735, 3.55134601120161229754577687887, 5.26450011311497926330441046561, 6.43072198561875065390956260194, 7.67547147761005193974373667047, 8.938151933757543249533337833208, 10.26129001058863420401737230093, 11.38294134970274625644757268489, 12.24987492607035480579598607204, 13.19968321021743691608912095768

Graph of the ZZ-function along the critical line