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Properties

Label 2-882-63.20-c1-0-21
Degree 22
Conductor 882882
Sign −0.118−0.992i-0.118 - 0.992i
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 + (1.28 + 1.16i)3-s + (0.499 + 0.866i)4-s + (0.450 + 0.779i)5-s + (0.533 + 1.64i)6-s + 0.999i·8-s + (0.308 + 2.98i)9-s + 0.900i·10-s + (2.70 + 1.56i)11-s + (−0.361 + 1.69i)12-s + (1.99 − 1.14i)13-s + (−0.325 + 1.52i)15-s + (−0.5 + 0.866i)16-s − 5.15·17-s + (−1.22 + 2.73i)18-s − 2.74i·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.742 + 0.669i)3-s + (0.249 + 0.433i)4-s + (0.201 + 0.348i)5-s + (0.217 + 0.672i)6-s + 0.353i·8-s + (0.102 + 0.994i)9-s + 0.284i·10-s + (0.816 + 0.471i)11-s + (−0.104 + 0.488i)12-s + (0.552 − 0.318i)13-s + (−0.0840 + 0.393i)15-s + (−0.125 + 0.216i)16-s − 1.24·17-s + (−0.288 + 0.645i)18-s − 0.630i·19-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((−0.118−0.992i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((−0.118−0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: −0.118−0.992i-0.118 - 0.992i
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), −0.118−0.992i)(2,\ 882,\ (\ :1/2),\ -0.118 - 0.992i)

Particular Values

L(1)L(1) ≈\approx 1.98238+2.23262i1.98238 + 2.23262i
L(12)L(\frac12) ≈\approx 1.98238+2.23262i1.98238 + 2.23262i
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+(−1.28−1.16i)T 1 + (-1.28 - 1.16i)T
7 1 1
good5 1+(−0.450−0.779i)T+(−2.5+4.33i)T2 1 + (-0.450 - 0.779i)T + (-2.5 + 4.33i)T^{2}
11 1+(−2.70−1.56i)T+(5.5+9.52i)T2 1 + (-2.70 - 1.56i)T + (5.5 + 9.52i)T^{2}
13 1+(−1.99+1.14i)T+(6.5−11.2i)T2 1 + (-1.99 + 1.14i)T + (6.5 - 11.2i)T^{2}
17 1+5.15T+17T2 1 + 5.15T + 17T^{2}
19 1+2.74iT−19T2 1 + 2.74iT - 19T^{2}
23 1+(−1.48+0.857i)T+(11.5−19.9i)T2 1 + (-1.48 + 0.857i)T + (11.5 - 19.9i)T^{2}
29 1+(1.85+1.07i)T+(14.5+25.1i)T2 1 + (1.85 + 1.07i)T + (14.5 + 25.1i)T^{2}
31 1+(8.66−5.00i)T+(15.5−26.8i)T2 1 + (8.66 - 5.00i)T + (15.5 - 26.8i)T^{2}
37 1−9.47T+37T2 1 - 9.47T + 37T^{2}
41 1+(1.22+2.11i)T+(−20.5+35.5i)T2 1 + (1.22 + 2.11i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.273−0.473i)T+(−21.5−37.2i)T2 1 + (0.273 - 0.473i)T + (-21.5 - 37.2i)T^{2}
47 1+(−3.93+6.80i)T+(−23.5−40.7i)T2 1 + (-3.93 + 6.80i)T + (-23.5 - 40.7i)T^{2}
53 1−13.9iT−53T2 1 - 13.9iT - 53T^{2}
59 1+(−3.99−6.91i)T+(−29.5+51.0i)T2 1 + (-3.99 - 6.91i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.28+3.62i)T+(30.5+52.8i)T2 1 + (6.28 + 3.62i)T + (30.5 + 52.8i)T^{2}
67 1+(1.83+3.17i)T+(−33.5+58.0i)T2 1 + (1.83 + 3.17i)T + (-33.5 + 58.0i)T^{2}
71 1+14.1iT−71T2 1 + 14.1iT - 71T^{2}
73 1+12.6iT−73T2 1 + 12.6iT - 73T^{2}
79 1+(−3.27+5.67i)T+(−39.5−68.4i)T2 1 + (-3.27 + 5.67i)T + (-39.5 - 68.4i)T^{2}
83 1+(0.184−0.319i)T+(−41.5−71.8i)T2 1 + (0.184 - 0.319i)T + (-41.5 - 71.8i)T^{2}
89 1−12.0T+89T2 1 - 12.0T + 89T^{2}
97 1+(8.86+5.12i)T+(48.5+84.0i)T2 1 + (8.86 + 5.12i)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

−10.54979448111170542079888210155, −9.166343697197955903279780914809, −8.946594268791188841466652933506, −7.73672884010261602497210765497, −6.91078767809104605324773131284, −6.03727930266735388680528326651, −4.82365412821909214883670031306, −4.13182025286398108780207731661, −3.12572417579885815315808578264, −2.07286702692552144011863249123, 1.21528612686041622708283754331, 2.23198702496750843319658778816, 3.50714590215879572041658242938, 4.22095205746676741332414947567, 5.61579400515637402175584616328, 6.43093986785098084165709478868, 7.21984145798884705521272357445, 8.350365018176095638400880270782, 9.109637500216336164515126672084, 9.666515070423246977106748866963

Graph of the ZZ-function along the critical line