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Properties

Label 2-567-7.4-c1-0-17
Degree 22
Conductor 567567
Sign −0.979−0.202i-0.979 - 0.202i
Analytic cond. 4.527514.52751
Root an. cond. 2.127792.12779
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  + (−1.29 − 2.24i)2-s + (−2.34 + 4.06i)4-s + (−1.14 − 1.97i)5-s + (2.45 − 0.989i)7-s + 6.97·8-s + (−2.95 + 5.11i)10-s + (1.47 − 2.56i)11-s + 4.26·13-s + (−5.39 − 4.21i)14-s + (−4.32 − 7.49i)16-s + (0.764 − 1.32i)17-s + (−3.69 − 6.39i)19-s + 10.7·20-s − 7.64·22-s + (3.07 + 5.32i)23-s + ⋯
L(s)  = 1  + (−0.914 − 1.58i)2-s + (−1.17 + 2.03i)4-s + (−0.510 − 0.884i)5-s + (0.927 − 0.374i)7-s + 2.46·8-s + (−0.934 + 1.61i)10-s + (0.445 − 0.771i)11-s + 1.18·13-s + (−1.44 − 1.12i)14-s + (−1.08 − 1.87i)16-s + (0.185 − 0.321i)17-s + (−0.846 − 1.46i)19-s + 2.39·20-s − 1.63·22-s + (0.641 + 1.11i)23-s + ⋯

Functional equation

Λ(s)=(567s/2 ΓC(s) L(s)=((−0.979−0.202i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(567s/2 ΓC(s+1/2) L(s)=((−0.979−0.202i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 567567    =    34⋅73^{4} \cdot 7
Sign: −0.979−0.202i-0.979 - 0.202i
Analytic conductor: 4.527514.52751
Root analytic conductor: 2.127792.12779
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ567(487,⋅)\chi_{567} (487, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 567, ( :1/2), −0.979−0.202i)(2,\ 567,\ (\ :1/2),\ -0.979 - 0.202i)

Particular Values

L(1)L(1) ≈\approx 0.0811394+0.794087i0.0811394 + 0.794087i
L(12)L(\frac12) ≈\approx 0.0811394+0.794087i0.0811394 + 0.794087i
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)
bad3 1 1
7 1+(−2.45+0.989i)T 1 + (-2.45 + 0.989i)T
good2 1+(1.29+2.24i)T+(−1+1.73i)T2 1 + (1.29 + 2.24i)T + (-1 + 1.73i)T^{2}
5 1+(1.14+1.97i)T+(−2.5+4.33i)T2 1 + (1.14 + 1.97i)T + (-2.5 + 4.33i)T^{2}
11 1+(−1.47+2.56i)T+(−5.5−9.52i)T2 1 + (-1.47 + 2.56i)T + (-5.5 - 9.52i)T^{2}
13 1−4.26T+13T2 1 - 4.26T + 13T^{2}
17 1+(−0.764+1.32i)T+(−8.5−14.7i)T2 1 + (-0.764 + 1.32i)T + (-8.5 - 14.7i)T^{2}
19 1+(3.69+6.39i)T+(−9.5+16.4i)T2 1 + (3.69 + 6.39i)T + (-9.5 + 16.4i)T^{2}
23 1+(−3.07−5.32i)T+(−11.5+19.9i)T2 1 + (-3.07 - 5.32i)T + (-11.5 + 19.9i)T^{2}
29 1+2.34T+29T2 1 + 2.34T + 29T^{2}
31 1+(3.11−5.38i)T+(−15.5−26.8i)T2 1 + (3.11 - 5.38i)T + (-15.5 - 26.8i)T^{2}
37 1+(3.58+6.21i)T+(−18.5+32.0i)T2 1 + (3.58 + 6.21i)T + (-18.5 + 32.0i)T^{2}
41 1+7.89T+41T2 1 + 7.89T + 41T^{2}
43 1−0.834T+43T2 1 - 0.834T + 43T^{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+(3.71−6.44i)T+(−26.5−45.8i)T2 1 + (3.71 - 6.44i)T + (-26.5 - 45.8i)T^{2}
59 1+(−2.31+4.00i)T+(−29.5−51.0i)T2 1 + (-2.31 + 4.00i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.56−6.17i)T+(−30.5+52.8i)T2 1 + (-3.56 - 6.17i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.66+2.87i)T+(−33.5−58.0i)T2 1 + (-1.66 + 2.87i)T + (-33.5 - 58.0i)T^{2}
71 1−0.160T+71T2 1 - 0.160T + 71T^{2}
73 1+(−0.190+0.329i)T+(−36.5−63.2i)T2 1 + (-0.190 + 0.329i)T + (-36.5 - 63.2i)T^{2}
79 1+(−3.97−6.88i)T+(−39.5+68.4i)T2 1 + (-3.97 - 6.88i)T + (-39.5 + 68.4i)T^{2}
83 1−4.29T+83T2 1 - 4.29T + 83T^{2}
89 1+(−3.02−5.24i)T+(−44.5+77.0i)T2 1 + (-3.02 - 5.24i)T + (-44.5 + 77.0i)T^{2}
97 1+1.32T+97T2 1 + 1.32T + 97T^{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

−10.60326633351614945220489364695, −9.247690744055262978854226996565, −8.713739201457723557986293555397, −8.230969955003856152059665021997, −7.07350210043332575035330976142, −5.20442330910704329902035238741, −4.15690507806590389127221330874, −3.32299023482943864578219838519, −1.67962559926462854111824903477, −0.69491116027861631629233463245, 1.64686653680827616898102988237, 3.85437564566033959001911822084, 5.04377372336568548520017354712, 6.17763829320418665201814393965, 6.74796466564479736261787165812, 7.78905575898734090438604795743, 8.289772088724781405544398818394, 9.094326454606083495736008949070, 10.23152877811182990524707837597, 10.84331966763799627996189629843

Graph of the ZZ-function along the critical line