Properties

Label 2-833-119.16-c1-0-55
Degree 22
Conductor 833833
Sign 0.03460.999i-0.0346 - 0.999i
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  + (1.15 − 2.00i)2-s + (0.115 − 0.0667i)3-s + (−1.68 − 2.92i)4-s + (−2.18 − 1.26i)5-s − 0.309i·6-s − 3.19·8-s + (−1.49 + 2.58i)9-s + (−5.07 + 2.92i)10-s + (−4.29 + 2.48i)11-s + (−0.390 − 0.225i)12-s − 2.86·13-s − 0.337·15-s + (−0.331 + 0.573i)16-s + (−3.19 + 2.60i)17-s + (3.45 + 5.99i)18-s + (2.98 − 5.17i)19-s + ⋯
L(s)  = 1  + (0.820 − 1.42i)2-s + (0.0667 − 0.0385i)3-s + (−0.844 − 1.46i)4-s + (−0.978 − 0.564i)5-s − 0.126i·6-s − 1.13·8-s + (−0.497 + 0.860i)9-s + (−1.60 + 0.926i)10-s + (−1.29 + 0.748i)11-s + (−0.112 − 0.0651i)12-s − 0.793·13-s − 0.0870·15-s + (−0.0828 + 0.143i)16-s + (−0.774 + 0.632i)17-s + (0.815 + 1.41i)18-s + (0.685 − 1.18i)19-s + ⋯

Functional equation

Λ(s)=(833s/2ΓC(s)L(s)=((0.03460.999i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0346 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(833s/2ΓC(s+1/2)L(s)=((0.03460.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0346 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 833833    =    72177^{2} \cdot 17
Sign: 0.03460.999i-0.0346 - 0.999i
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), 0.03460.999i)(2,\ 833,\ (\ :1/2),\ -0.0346 - 0.999i)

Particular Values

L(1)L(1) \approx 0.304477+0.315228i0.304477 + 0.315228i
L(12)L(\frac12) \approx 0.304477+0.315228i0.304477 + 0.315228i
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)
bad7 1 1
17 1+(3.192.60i)T 1 + (3.19 - 2.60i)T
good2 1+(1.15+2.00i)T+(11.73i)T2 1 + (-1.15 + 2.00i)T + (-1 - 1.73i)T^{2}
3 1+(0.115+0.0667i)T+(1.52.59i)T2 1 + (-0.115 + 0.0667i)T + (1.5 - 2.59i)T^{2}
5 1+(2.18+1.26i)T+(2.5+4.33i)T2 1 + (2.18 + 1.26i)T + (2.5 + 4.33i)T^{2}
11 1+(4.292.48i)T+(5.59.52i)T2 1 + (4.29 - 2.48i)T + (5.5 - 9.52i)T^{2}
13 1+2.86T+13T2 1 + 2.86T + 13T^{2}
19 1+(2.98+5.17i)T+(9.516.4i)T2 1 + (-2.98 + 5.17i)T + (-9.5 - 16.4i)T^{2}
23 1+(2.591.49i)T+(11.5+19.9i)T2 1 + (-2.59 - 1.49i)T + (11.5 + 19.9i)T^{2}
29 1+4.37iT29T2 1 + 4.37iT - 29T^{2}
31 1+(2.331.35i)T+(15.526.8i)T2 1 + (2.33 - 1.35i)T + (15.5 - 26.8i)T^{2}
37 1+(5.41+3.12i)T+(18.5+32.0i)T2 1 + (5.41 + 3.12i)T + (18.5 + 32.0i)T^{2}
41 1+6.74iT41T2 1 + 6.74iT - 41T^{2}
43 1+6.21T+43T2 1 + 6.21T + 43T^{2}
47 1+(0.7881.36i)T+(23.540.7i)T2 1 + (0.788 - 1.36i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.09870.171i)T+(26.5+45.8i)T2 1 + (-0.0987 - 0.171i)T + (-26.5 + 45.8i)T^{2}
59 1+(4.067.04i)T+(29.5+51.0i)T2 1 + (-4.06 - 7.04i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.4720.272i)T+(30.5+52.8i)T2 1 + (-0.472 - 0.272i)T + (30.5 + 52.8i)T^{2}
67 1+(0.358+0.620i)T+(33.5+58.0i)T2 1 + (0.358 + 0.620i)T + (-33.5 + 58.0i)T^{2}
71 1+10.8iT71T2 1 + 10.8iT - 71T^{2}
73 1+(4.66+2.69i)T+(36.563.2i)T2 1 + (-4.66 + 2.69i)T + (36.5 - 63.2i)T^{2}
79 1+(12.06.95i)T+(39.5+68.4i)T2 1 + (-12.0 - 6.95i)T + (39.5 + 68.4i)T^{2}
83 1+10.7T+83T2 1 + 10.7T + 83T^{2}
89 1+(8.4814.7i)T+(44.577.0i)T2 1 + (8.48 - 14.7i)T + (-44.5 - 77.0i)T^{2}
97 1+5.73iT97T2 1 + 5.73iT - 97T^{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

−9.928765097665671839747476066593, −8.897872461020180742175418615087, −7.903207643878317620976985868819, −7.19000576250155887900002754535, −5.28794098626800608744054381893, −4.94443298387792135733673451021, −4.05530200738932691228740310494, −2.83717653740312175916028321825, −2.05815181510093366959111780404, −0.15197690145134078323907594760, 3.00245931137684906308100495780, 3.64050583450964964075349361215, 4.88132078002752064541744543477, 5.56588816500957629652754692378, 6.62065157487354341303826817708, 7.26779959437147901851550264371, 8.026076113010620730789115202761, 8.658057609869900541223110187208, 9.913863209820265767993395371802, 11.00881798078988051217537918319

Graph of the ZZ-function along the critical line