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Properties

Label 2-882-63.20-c1-0-6
Degree 22
Conductor 882882
Sign −0.380−0.924i-0.380 - 0.924i
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  + (0.866 + 0.5i)2-s + (−1.43 − 0.974i)3-s + (0.499 + 0.866i)4-s + (1.80 + 3.13i)5-s + (−0.752 − 1.56i)6-s + 0.999i·8-s + (1.09 + 2.79i)9-s + 3.61i·10-s + (1.73 + 1.00i)11-s + (0.128 − 1.72i)12-s + (−2.95 + 1.70i)13-s + (0.465 − 6.25i)15-s + (−0.5 + 0.866i)16-s − 6.17·17-s + (−0.443 + 2.96i)18-s − 1.01i·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.826 − 0.562i)3-s + (0.249 + 0.433i)4-s + (0.809 + 1.40i)5-s + (−0.307 − 0.636i)6-s + 0.353i·8-s + (0.366 + 0.930i)9-s + 1.14i·10-s + (0.523 + 0.302i)11-s + (0.0370 − 0.498i)12-s + (−0.818 + 0.472i)13-s + (0.120 − 1.61i)15-s + (−0.125 + 0.216i)16-s − 1.49·17-s + (−0.104 + 0.699i)18-s − 0.232i·19-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((−0.380−0.924i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((−0.380−0.924i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: −0.380−0.924i-0.380 - 0.924i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(587,⋅)\chi_{882} (587, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), −0.380−0.924i)(2,\ 882,\ (\ :1/2),\ -0.380 - 0.924i)

Particular Values

L(1)L(1) ≈\approx 0.927027+1.38436i0.927027 + 1.38436i
L(12)L(\frac12) ≈\approx 0.927027+1.38436i0.927027 + 1.38436i
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
3 1+(1.43+0.974i)T 1 + (1.43 + 0.974i)T
7 1 1
good5 1+(−1.80−3.13i)T+(−2.5+4.33i)T2 1 + (-1.80 - 3.13i)T + (-2.5 + 4.33i)T^{2}
11 1+(−1.73−1.00i)T+(5.5+9.52i)T2 1 + (-1.73 - 1.00i)T + (5.5 + 9.52i)T^{2}
13 1+(2.95−1.70i)T+(6.5−11.2i)T2 1 + (2.95 - 1.70i)T + (6.5 - 11.2i)T^{2}
17 1+6.17T+17T2 1 + 6.17T + 17T^{2}
19 1+1.01iT−19T2 1 + 1.01iT - 19T^{2}
23 1+(−2.62+1.51i)T+(11.5−19.9i)T2 1 + (-2.62 + 1.51i)T + (11.5 - 19.9i)T^{2}
29 1+(−5.04−2.91i)T+(14.5+25.1i)T2 1 + (-5.04 - 2.91i)T + (14.5 + 25.1i)T^{2}
31 1+(0.787−0.454i)T+(15.5−26.8i)T2 1 + (0.787 - 0.454i)T + (15.5 - 26.8i)T^{2}
37 1+7.33T+37T2 1 + 7.33T + 37T^{2}
41 1+(−2.85−4.93i)T+(−20.5+35.5i)T2 1 + (-2.85 - 4.93i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.39−4.15i)T+(−21.5−37.2i)T2 1 + (2.39 - 4.15i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.11−1.93i)T+(−23.5−40.7i)T2 1 + (1.11 - 1.93i)T + (-23.5 - 40.7i)T^{2}
53 1+8.75iT−53T2 1 + 8.75iT - 53T^{2}
59 1+(−4.49−7.78i)T+(−29.5+51.0i)T2 1 + (-4.49 - 7.78i)T + (-29.5 + 51.0i)T^{2}
61 1+(−12.7−7.35i)T+(30.5+52.8i)T2 1 + (-12.7 - 7.35i)T + (30.5 + 52.8i)T^{2}
67 1+(−4.15−7.20i)T+(−33.5+58.0i)T2 1 + (-4.15 - 7.20i)T + (-33.5 + 58.0i)T^{2}
71 1+0.466iT−71T2 1 + 0.466iT - 71T^{2}
73 1+4.21iT−73T2 1 + 4.21iT - 73T^{2}
79 1+(1.91−3.31i)T+(−39.5−68.4i)T2 1 + (1.91 - 3.31i)T + (-39.5 - 68.4i)T^{2}
83 1+(−4.00+6.93i)T+(−41.5−71.8i)T2 1 + (-4.00 + 6.93i)T + (-41.5 - 71.8i)T^{2}
89 1+4.78T+89T2 1 + 4.78T + 89T^{2}
97 1+(−10.1−5.87i)T+(48.5+84.0i)T2 1 + (-10.1 - 5.87i)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

−10.57751045733297337845414752165, −9.798696678894867559059629135043, −8.617244276082340623668369846115, −7.19488071405138763785813960169, −6.82566158608493545443465218754, −6.34600139761686583724180822589, −5.25725095354072948809471964427, −4.36992601418983523471842980006, −2.81252123522069729099111091976, −1.95015782165380097366666035260, 0.70851202143179514066166979933, 2.09869144020704641469760301887, 3.75098772698276330874985606113, 4.76388454176543541952920592719, 5.19913988176945092250309426266, 6.07853671439369510104895434598, 6.93038622058270277643828907729, 8.538916191966004959357691931868, 9.247059390313089784074567353015, 9.929699104273912930654693215417

Graph of the ZZ-function along the critical line