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Properties

Label 2-882-63.4-c1-0-27
Degree 22
Conductor 882882
Sign −0.608+0.793i-0.608 + 0.793i
Analytic cond. 7.042807.04280
Root an. cond. 2.653822.65382
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  + (0.5 − 0.866i)2-s + (−0.796 + 1.53i)3-s + (−0.499 − 0.866i)4-s − 0.593·5-s + (0.933 + 1.45i)6-s − 0.999·8-s + (−1.73 − 2.45i)9-s + (−0.296 + 0.514i)10-s − 0.593·11-s + (1.73 − 0.0789i)12-s + (1.25 − 2.17i)13-s + (0.472 − 0.912i)15-s + (−0.5 + 0.866i)16-s + (−1.46 + 2.52i)17-s + (−2.98 + 0.273i)18-s + (−2.69 − 4.66i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.460 + 0.887i)3-s + (−0.249 − 0.433i)4-s − 0.265·5-s + (0.381 + 0.595i)6-s − 0.353·8-s + (−0.576 − 0.816i)9-s + (−0.0938 + 0.162i)10-s − 0.178·11-s + (0.499 − 0.0227i)12-s + (0.348 − 0.603i)13-s + (0.122 − 0.235i)15-s + (−0.125 + 0.216i)16-s + (−0.354 + 0.613i)17-s + (−0.704 + 0.0643i)18-s + (−0.617 − 1.06i)19-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((−0.608+0.793i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((−0.608+0.793i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: −0.608+0.793i-0.608 + 0.793i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(67,⋅)\chi_{882} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), −0.608+0.793i)(2,\ 882,\ (\ :1/2),\ -0.608 + 0.793i)

Particular Values

L(1)L(1) ≈\approx 0.352078−0.714185i0.352078 - 0.714185i
L(12)L(\frac12) ≈\approx 0.352078−0.714185i0.352078 - 0.714185i
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1+(0.796−1.53i)T 1 + (0.796 - 1.53i)T
7 1 1
good5 1+0.593T+5T2 1 + 0.593T + 5T^{2}
11 1+0.593T+11T2 1 + 0.593T + 11T^{2}
13 1+(−1.25+2.17i)T+(−6.5−11.2i)T2 1 + (-1.25 + 2.17i)T + (-6.5 - 11.2i)T^{2}
17 1+(1.46−2.52i)T+(−8.5−14.7i)T2 1 + (1.46 - 2.52i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.69+4.66i)T+(−9.5+16.4i)T2 1 + (2.69 + 4.66i)T + (-9.5 + 16.4i)T^{2}
23 1−4.46T+23T2 1 - 4.46T + 23T^{2}
29 1+(3.09+5.36i)T+(−14.5+25.1i)T2 1 + (3.09 + 5.36i)T + (-14.5 + 25.1i)T^{2}
31 1+(3.93+6.81i)T+(−15.5+26.8i)T2 1 + (3.93 + 6.81i)T + (-15.5 + 26.8i)T^{2}
37 1+(−0.5−0.866i)T+(−18.5+32.0i)T2 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2}
41 1+(−0.136+0.236i)T+(−20.5−35.5i)T2 1 + (-0.136 + 0.236i)T + (-20.5 - 35.5i)T^{2}
43 1+(5.58+9.66i)T+(−21.5+37.2i)T2 1 + (5.58 + 9.66i)T + (-21.5 + 37.2i)T^{2}
47 1+(−6.08+10.5i)T+(−23.5−40.7i)T2 1 + (-6.08 + 10.5i)T + (-23.5 - 40.7i)T^{2}
53 1+(−4.02+6.97i)T+(−26.5−45.8i)T2 1 + (-4.02 + 6.97i)T + (-26.5 - 45.8i)T^{2}
59 1+(−4.32−7.48i)T+(−29.5+51.0i)T2 1 + (-4.32 - 7.48i)T + (-29.5 + 51.0i)T^{2}
61 1+(3.32−5.75i)T+(−30.5−52.8i)T2 1 + (3.32 - 5.75i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.956−1.65i)T+(−33.5+58.0i)T2 1 + (-0.956 - 1.65i)T + (-33.5 + 58.0i)T^{2}
71 1+14.4T+71T2 1 + 14.4T + 71T^{2}
73 1+(3.95−6.85i)T+(−36.5−63.2i)T2 1 + (3.95 - 6.85i)T + (-36.5 - 63.2i)T^{2}
79 1+(−4.62+8.00i)T+(−39.5−68.4i)T2 1 + (-4.62 + 8.00i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.85+6.66i)T+(−41.5+71.8i)T2 1 + (3.85 + 6.66i)T + (-41.5 + 71.8i)T^{2}
89 1+(−6.21−10.7i)T+(−44.5+77.0i)T2 1 + (-6.21 - 10.7i)T + (-44.5 + 77.0i)T^{2}
97 1+(5.86+10.1i)T+(−48.5+84.0i)T2 1 + (5.86 + 10.1i)T + (-48.5 + 84.0i)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

−10.14570812906514688203649201206, −9.128727702070625757448230636540, −8.490700742993706447064813578370, −7.17504324354927848316463890733, −6.00426852614445019466856811867, −5.33008030029587114905917861928, −4.26082429886164228258604970531, −3.62640808473415162299507896563, −2.35122072651991706054154157876, −0.35676202302856068109599730474, 1.60982015257796281115018729157, 3.09536346883604719817534817938, 4.37771959156437190318007914439, 5.35354772291003960569278005938, 6.20104605944529926894323329318, 6.98389898657521094439239212326, 7.66855486851094427028996958333, 8.511365369229489965549495203750, 9.352605337987990635341621962292, 10.71903823938550972077457143256

Graph of the ZZ-function along the critical line