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Properties

Label 2-882-63.41-c1-0-14
Degree 22
Conductor 882882
Sign 0.701−0.712i0.701 - 0.712i
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 + (1.59 − 0.681i)3-s + (0.499 − 0.866i)4-s + (−0.714 + 1.23i)5-s + (−1.03 + 1.38i)6-s + 0.999i·8-s + (2.07 − 2.17i)9-s − 1.42i·10-s + (−2.96 + 1.70i)11-s + (0.206 − 1.71i)12-s + (5.48 + 3.16i)13-s + (−0.294 + 2.45i)15-s + (−0.5 − 0.866i)16-s − 2.28·17-s + (−0.708 + 2.91i)18-s − 2.16i·19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.919 − 0.393i)3-s + (0.249 − 0.433i)4-s + (−0.319 + 0.553i)5-s + (−0.423 + 0.565i)6-s + 0.353i·8-s + (0.690 − 0.723i)9-s − 0.452i·10-s + (−0.892 + 0.515i)11-s + (0.0594 − 0.496i)12-s + (1.52 + 0.878i)13-s + (−0.0760 + 0.634i)15-s + (−0.125 − 0.216i)16-s − 0.553·17-s + (−0.167 + 0.687i)18-s − 0.497i·19-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((0.701−0.712i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((0.701−0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 0.701−0.712i0.701 - 0.712i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(293,⋅)\chi_{882} (293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), 0.701−0.712i)(2,\ 882,\ (\ :1/2),\ 0.701 - 0.712i)

Particular Values

L(1)L(1) ≈\approx 1.42434+0.596405i1.42434 + 0.596405i
L(12)L(\frac12) ≈\approx 1.42434+0.596405i1.42434 + 0.596405i
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+(−1.59+0.681i)T 1 + (-1.59 + 0.681i)T
7 1 1
good5 1+(0.714−1.23i)T+(−2.5−4.33i)T2 1 + (0.714 - 1.23i)T + (-2.5 - 4.33i)T^{2}
11 1+(2.96−1.70i)T+(5.5−9.52i)T2 1 + (2.96 - 1.70i)T + (5.5 - 9.52i)T^{2}
13 1+(−5.48−3.16i)T+(6.5+11.2i)T2 1 + (-5.48 - 3.16i)T + (6.5 + 11.2i)T^{2}
17 1+2.28T+17T2 1 + 2.28T + 17T^{2}
19 1+2.16iT−19T2 1 + 2.16iT - 19T^{2}
23 1+(−6.97−4.02i)T+(11.5+19.9i)T2 1 + (-6.97 - 4.02i)T + (11.5 + 19.9i)T^{2}
29 1+(0.298−0.172i)T+(14.5−25.1i)T2 1 + (0.298 - 0.172i)T + (14.5 - 25.1i)T^{2}
31 1+(−3.76−2.17i)T+(15.5+26.8i)T2 1 + (-3.76 - 2.17i)T + (15.5 + 26.8i)T^{2}
37 1+2.15T+37T2 1 + 2.15T + 37T^{2}
41 1+(0.202−0.350i)T+(−20.5−35.5i)T2 1 + (0.202 - 0.350i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.90−5.03i)T+(−21.5+37.2i)T2 1 + (-2.90 - 5.03i)T + (-21.5 + 37.2i)T^{2}
47 1+(2.75+4.77i)T+(−23.5+40.7i)T2 1 + (2.75 + 4.77i)T + (-23.5 + 40.7i)T^{2}
53 1−9.88iT−53T2 1 - 9.88iT - 53T^{2}
59 1+(−5.51+9.55i)T+(−29.5−51.0i)T2 1 + (-5.51 + 9.55i)T + (-29.5 - 51.0i)T^{2}
61 1+(−9.94+5.73i)T+(30.5−52.8i)T2 1 + (-9.94 + 5.73i)T + (30.5 - 52.8i)T^{2}
67 1+(2.12−3.68i)T+(−33.5−58.0i)T2 1 + (2.12 - 3.68i)T + (-33.5 - 58.0i)T^{2}
71 1−3.55iT−71T2 1 - 3.55iT - 71T^{2}
73 1−0.232iT−73T2 1 - 0.232iT - 73T^{2}
79 1+(7.28+12.6i)T+(−39.5+68.4i)T2 1 + (7.28 + 12.6i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.811+1.40i)T+(−41.5+71.8i)T2 1 + (0.811 + 1.40i)T + (-41.5 + 71.8i)T^{2}
89 1+4.05T+89T2 1 + 4.05T + 89T^{2}
97 1+(−9.18+5.30i)T+(48.5−84.0i)T2 1 + (-9.18 + 5.30i)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

−10.08731546398990067921788562140, −9.090381097387914821948537628842, −8.660201998585122885566425733189, −7.66884948294345094664576691074, −7.01765453944542912593552962324, −6.37709989907637661183344804455, −4.94131973614177076907902370178, −3.65919003968874066784519450425, −2.64633661360778512235650482947, −1.37925347969871100902780228075, 0.942276407603216963042631486671, 2.53264813417342685814344712667, 3.41611790471131309462250585582, 4.41332599357883119025763259328, 5.54993361167998594101170745171, 6.86005304704800607968829178123, 8.019009843080315113505161066080, 8.464480588020801993616141751452, 8.916138744031926907253028481786, 10.09893345862168230760005229979

Graph of the ZZ-function along the critical line