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Properties

Label 2-882-9.4-c1-0-31
Degree 22
Conductor 882882
Sign 0.999+0.0334i0.999 + 0.0334i
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 + (−0.619 + 1.61i)3-s + (−0.499 + 0.866i)4-s + (1.59 − 2.75i)5-s + (−1.71 + 0.272i)6-s − 0.999·8-s + (−2.23 − 2.00i)9-s + 3.18·10-s + (−1.59 − 2.75i)11-s + (−1.09 − 1.34i)12-s + (2.85 − 4.93i)13-s + (3.47 + 4.28i)15-s + (−0.5 − 0.866i)16-s + 1.52·17-s + (0.619 − 2.93i)18-s + 1.28·19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.357 + 0.933i)3-s + (−0.249 + 0.433i)4-s + (0.711 − 1.23i)5-s + (−0.698 + 0.111i)6-s − 0.353·8-s + (−0.744 − 0.668i)9-s + 1.00·10-s + (−0.479 − 0.830i)11-s + (−0.314 − 0.388i)12-s + (0.790 − 1.36i)13-s + (0.896 + 1.10i)15-s + (−0.125 − 0.216i)16-s + 0.369·17-s + (0.146 − 0.691i)18-s + 0.294·19-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((0.999+0.0334i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0334i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((0.999+0.0334i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0334i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 0.999+0.0334i0.999 + 0.0334i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(589,⋅)\chi_{882} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), 0.999+0.0334i)(2,\ 882,\ (\ :1/2),\ 0.999 + 0.0334i)

Particular Values

L(1)L(1) ≈\approx 1.64054−0.0274690i1.64054 - 0.0274690i
L(12)L(\frac12) ≈\approx 1.64054−0.0274690i1.64054 - 0.0274690i
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+(0.619−1.61i)T 1 + (0.619 - 1.61i)T
7 1 1
good5 1+(−1.59+2.75i)T+(−2.5−4.33i)T2 1 + (-1.59 + 2.75i)T + (-2.5 - 4.33i)T^{2}
11 1+(1.59+2.75i)T+(−5.5+9.52i)T2 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2}
13 1+(−2.85+4.93i)T+(−6.5−11.2i)T2 1 + (-2.85 + 4.93i)T + (-6.5 - 11.2i)T^{2}
17 1−1.52T+17T2 1 - 1.52T + 17T^{2}
19 1−1.28T+19T2 1 - 1.28T + 19T^{2}
23 1+(1.11−1.93i)T+(−11.5−19.9i)T2 1 + (1.11 - 1.93i)T + (-11.5 - 19.9i)T^{2}
29 1+(3.54+6.13i)T+(−14.5+25.1i)T2 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2}
31 1+(−4.71+8.15i)T+(−15.5−26.8i)T2 1 + (-4.71 + 8.15i)T + (-15.5 - 26.8i)T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1+(2.80−4.85i)T+(−20.5−35.5i)T2 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.41−5.91i)T+(−21.5+37.2i)T2 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2}
47 1+(−2.91−5.04i)T+(−23.5+40.7i)T2 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2}
53 1+2.05T+53T2 1 + 2.05T + 53T^{2}
59 1+(−0.562+0.974i)T+(−29.5−51.0i)T2 1 + (-0.562 + 0.974i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.56+2.70i)T+(−30.5+52.8i)T2 1 + (1.56 + 2.70i)T + (-30.5 + 52.8i)T^{2}
67 1+(5.48−9.49i)T+(−33.5−58.0i)T2 1 + (5.48 - 9.49i)T + (-33.5 - 58.0i)T^{2}
71 1−8.69T+71T2 1 - 8.69T + 71T^{2}
73 1−4.96T+73T2 1 - 4.96T + 73T^{2}
79 1+(−2.06−3.58i)T+(−39.5+68.4i)T2 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2}
83 1+(4.03+6.98i)T+(−41.5+71.8i)T2 1 + (4.03 + 6.98i)T + (-41.5 + 71.8i)T^{2}
89 1+0.225T+89T2 1 + 0.225T + 89T^{2}
97 1+(−7.42−12.8i)T+(−48.5+84.0i)T2 1 + (-7.42 - 12.8i)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

−9.893037140678974178999666177947, −9.372111570723994803297917025874, −8.346972004559955260556932598624, −7.917761781804951435000711570644, −6.01915913539053626085803436887, −5.80676516797965043836080470921, −5.02841739944039028659275330582, −4.04831195630895217181137727373, −2.96027653085674315588026649525, −0.76691125660843399200042336221, 1.60752441804350838460588554488, 2.37914616324586324900511781359, 3.48578198654754047254959318870, 4.92868627411304202468032820975, 5.87787895924820000099887063422, 6.76615867646802994682505534078, 7.16268394521650274742897968641, 8.513972313878062558754640963757, 9.493704905343100460333267478768, 10.58466747969075770155463034181

Graph of the ZZ-function along the critical line