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Properties

Label 2-833-119.16-c1-0-16
Degree 22
Conductor 833833
Sign −0.968−0.249i-0.968 - 0.249i
Analytic cond. 6.651536.65153
Root an. cond. 2.579052.57905
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  + (−1.22 + 2.12i)2-s + (1.01 − 0.583i)3-s + (−2.00 − 3.47i)4-s + (2.29 + 1.32i)5-s + 2.86i·6-s + 4.94·8-s + (−0.819 + 1.41i)9-s + (−5.62 + 3.24i)10-s + (−3.03 + 1.75i)11-s + (−4.05 − 2.34i)12-s + 1.85·13-s + 3.09·15-s + (−2.04 + 3.54i)16-s + (1.59 + 3.80i)17-s + (−2.00 − 3.47i)18-s + (0.224 − 0.389i)19-s + ⋯
L(s)  = 1  + (−0.867 + 1.50i)2-s + (0.583 − 0.336i)3-s + (−1.00 − 1.73i)4-s + (1.02 + 0.592i)5-s + 1.16i·6-s + 1.74·8-s + (−0.273 + 0.472i)9-s + (−1.77 + 1.02i)10-s + (−0.915 + 0.528i)11-s + (−1.17 − 0.676i)12-s + 0.513·13-s + 0.798·15-s + (−0.511 + 0.885i)16-s + (0.387 + 0.921i)17-s + (−0.473 − 0.820i)18-s + (0.0515 − 0.0893i)19-s + ⋯

Functional equation

Λ(s)=(833s/2 ΓC(s) L(s)=((−0.968−0.249i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(833s/2 ΓC(s+1/2) L(s)=((−0.968−0.249i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 833833    =    72⋅177^{2} \cdot 17
Sign: −0.968−0.249i-0.968 - 0.249i
Analytic conductor: 6.651536.65153
Root analytic conductor: 2.579052.57905
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ833(373,⋅)\chi_{833} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 833, ( :1/2), −0.968−0.249i)(2,\ 833,\ (\ :1/2),\ -0.968 - 0.249i)

Particular Values

L(1)L(1) ≈\approx 0.140806+1.11145i0.140806 + 1.11145i
L(12)L(\frac12) ≈\approx 0.140806+1.11145i0.140806 + 1.11145i
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)
bad7 1 1
17 1+(−1.59−3.80i)T 1 + (-1.59 - 3.80i)T
good2 1+(1.22−2.12i)T+(−1−1.73i)T2 1 + (1.22 - 2.12i)T + (-1 - 1.73i)T^{2}
3 1+(−1.01+0.583i)T+(1.5−2.59i)T2 1 + (-1.01 + 0.583i)T + (1.5 - 2.59i)T^{2}
5 1+(−2.29−1.32i)T+(2.5+4.33i)T2 1 + (-2.29 - 1.32i)T + (2.5 + 4.33i)T^{2}
11 1+(3.03−1.75i)T+(5.5−9.52i)T2 1 + (3.03 - 1.75i)T + (5.5 - 9.52i)T^{2}
13 1−1.85T+13T2 1 - 1.85T + 13T^{2}
19 1+(−0.224+0.389i)T+(−9.5−16.4i)T2 1 + (-0.224 + 0.389i)T + (-9.5 - 16.4i)T^{2}
23 1+(4.18+2.41i)T+(11.5+19.9i)T2 1 + (4.18 + 2.41i)T + (11.5 + 19.9i)T^{2}
29 1−8.72iT−29T2 1 - 8.72iT - 29T^{2}
31 1+(−5.78+3.33i)T+(15.5−26.8i)T2 1 + (-5.78 + 3.33i)T + (15.5 - 26.8i)T^{2}
37 1+(−4.47−2.58i)T+(18.5+32.0i)T2 1 + (-4.47 - 2.58i)T + (18.5 + 32.0i)T^{2}
41 1+3.10iT−41T2 1 + 3.10iT - 41T^{2}
43 1+7.42T+43T2 1 + 7.42T + 43T^{2}
47 1+(6.16−10.6i)T+(−23.5−40.7i)T2 1 + (6.16 - 10.6i)T + (-23.5 - 40.7i)T^{2}
53 1+(−5.15−8.93i)T+(−26.5+45.8i)T2 1 + (-5.15 - 8.93i)T + (-26.5 + 45.8i)T^{2}
59 1+(−1.29−2.25i)T+(−29.5+51.0i)T2 1 + (-1.29 - 2.25i)T + (-29.5 + 51.0i)T^{2}
61 1+(3.27+1.89i)T+(30.5+52.8i)T2 1 + (3.27 + 1.89i)T + (30.5 + 52.8i)T^{2}
67 1+(−1.03−1.79i)T+(−33.5+58.0i)T2 1 + (-1.03 - 1.79i)T + (-33.5 + 58.0i)T^{2}
71 1+12.3iT−71T2 1 + 12.3iT - 71T^{2}
73 1+(1.95−1.13i)T+(36.5−63.2i)T2 1 + (1.95 - 1.13i)T + (36.5 - 63.2i)T^{2}
79 1+(−0.384−0.222i)T+(39.5+68.4i)T2 1 + (-0.384 - 0.222i)T + (39.5 + 68.4i)T^{2}
83 1−8.97T+83T2 1 - 8.97T + 83T^{2}
89 1+(0.598−1.03i)T+(−44.5−77.0i)T2 1 + (0.598 - 1.03i)T + (-44.5 - 77.0i)T^{2}
97 1+2.92iT−97T2 1 + 2.92iT - 97T^{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

−10.25976411099804227123966889105, −9.593851314693043445639324072416, −8.581418842769608506942228582413, −8.049750464979920104516144606584, −7.32324747378835406357169614396, −6.35212969998112927627036482801, −5.81203514352400168965344519685, −4.79821869181507408143423135345, −2.89028238720390634914543786873, −1.69425543117795749690652771006, 0.68429425803735148937868929958, 2.06185455331027516449550971897, 2.94366999547392587569030739118, 3.84896386343126094848873796920, 5.19710156390600463037232169639, 6.22347900447406404253488826295, 7.962894388375689185403270616931, 8.453866928849739138619978395233, 9.273544525024706444598845558749, 9.906051542980499876319182456818

Graph of the ZZ-function along the critical line