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Properties

Label 2-882-7.2-c3-0-27
Degree 22
Conductor 882882
Sign 0.991+0.126i0.991 + 0.126i
Analytic cond. 52.039652.0396
Root an. cond. 7.213857.21385
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−3.5 + 6.06i)5-s + 7.99·8-s + (−7 − 12.1i)10-s + (17.5 + 30.3i)11-s − 66·13-s + (−8 + 13.8i)16-s + (−29.5 − 51.0i)17-s + (68.5 − 118. i)19-s + 28·20-s − 70·22-s + (−3.5 + 6.06i)23-s + (38 + 65.8i)25-s + (66 − 114. i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.313 + 0.542i)5-s + 0.353·8-s + (−0.221 − 0.383i)10-s + (0.479 + 0.830i)11-s − 1.40·13-s + (−0.125 + 0.216i)16-s + (−0.420 − 0.728i)17-s + (0.827 − 1.43i)19-s + 0.313·20-s − 0.678·22-s + (−0.0317 + 0.0549i)23-s + (0.303 + 0.526i)25-s + (0.497 − 0.862i)26-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((0.991+0.126i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+3/2) L(s)=((0.991+0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 0.991+0.126i0.991 + 0.126i
Analytic conductor: 52.039652.0396
Root analytic conductor: 7.213857.21385
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ882(667,⋅)\chi_{882} (667, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :3/2), 0.991+0.126i)(2,\ 882,\ (\ :3/2),\ 0.991 + 0.126i)

Particular Values

L(2)L(2) ≈\approx 1.0229006091.022900609
L(12)L(\frac12) ≈\approx 1.0229006091.022900609
L(52)L(\frac{5}{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+(1−1.73i)T 1 + (1 - 1.73i)T
3 1 1
7 1 1
good5 1+(3.5−6.06i)T+(−62.5−108.i)T2 1 + (3.5 - 6.06i)T + (-62.5 - 108. i)T^{2}
11 1+(−17.5−30.3i)T+(−665.5+1.15e3i)T2 1 + (-17.5 - 30.3i)T + (-665.5 + 1.15e3i)T^{2}
13 1+66T+2.19e3T2 1 + 66T + 2.19e3T^{2}
17 1+(29.5+51.0i)T+(−2.45e3+4.25e3i)T2 1 + (29.5 + 51.0i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(−68.5+118.i)T+(−3.42e3−5.94e3i)T2 1 + (-68.5 + 118. i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(3.5−6.06i)T+(−6.08e3−1.05e4i)T2 1 + (3.5 - 6.06i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+106T+2.43e4T2 1 + 106T + 2.43e4T^{2}
31 1+(−37.5−64.9i)T+(−1.48e4+2.57e4i)T2 1 + (-37.5 - 64.9i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(5.5−9.52i)T+(−2.53e4−4.38e4i)T2 1 + (5.5 - 9.52i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+498T+6.89e4T2 1 + 498T + 6.89e4T^{2}
43 1−260T+7.95e4T2 1 - 260T + 7.95e4T^{2}
47 1+(−85.5+148.i)T+(−5.19e4−8.99e4i)T2 1 + (-85.5 + 148. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(208.5+361.i)T+(−7.44e4+1.28e5i)T2 1 + (208.5 + 361. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−8.5−14.7i)T+(−1.02e5+1.77e5i)T2 1 + (-8.5 - 14.7i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−25.5+44.1i)T+(−1.13e5−1.96e5i)T2 1 + (-25.5 + 44.1i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(219.5+380.i)T+(−1.50e5+2.60e5i)T2 1 + (219.5 + 380. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−784T+3.57e5T2 1 - 784T + 3.57e5T^{2}
73 1+(−147.5−255.i)T+(−1.94e5+3.36e5i)T2 1 + (-147.5 - 255. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(−247.5+428.i)T+(−2.46e5−4.26e5i)T2 1 + (-247.5 + 428. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1−932T+5.71e5T2 1 - 932T + 5.71e5T^{2}
89 1+(−436.5+756.i)T+(−3.52e5−6.10e5i)T2 1 + (-436.5 + 756. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1−290T+9.12e5T2 1 - 290T + 9.12e5T^{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

−9.520398906676197926588163413478, −9.076260019416269867917586082603, −7.78204159044455625372466350830, −7.07178880730073154365133204922, −6.75390374056931334616334687400, −5.21205112577994915283732849663, −4.67988562440756188989188847818, −3.25449409154516855377114838133, −2.07252921039306952139782110772, −0.40338328084342301524641908582, 0.832043311305833913523232234021, 2.03693154474428314659159210349, 3.32198324117737817605549476503, 4.21354458573971250352090719392, 5.23451480780146545295883837705, 6.30000082153689471637851609023, 7.51927821940859444470299721761, 8.187452947395834703355436028831, 8.997042314847306825319242943007, 9.783844137183922726823207024269

Graph of the ZZ-function along the critical line