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Properties

Label 2-936-104.101-c1-0-60
Degree 22
Conductor 936936
Sign 0.00641+0.999i0.00641 + 0.999i
Analytic cond. 7.473997.47399
Root an. cond. 2.733862.73386
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  + (1.36 − 0.366i)2-s + (1.73 − i)4-s − 0.267·5-s + (−3 − 1.73i)7-s + (1.99 − 2i)8-s + (−0.366 + 0.0980i)10-s + (−1 − 1.73i)11-s + (2.59 + 2.5i)13-s + (−4.73 − 1.26i)14-s + (1.99 − 3.46i)16-s + (3.23 − 5.59i)17-s + (2.36 − 4.09i)19-s + (−0.464 + 0.267i)20-s + (−2 − 1.99i)22-s + (−1.09 − 1.90i)23-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s − 0.119·5-s + (−1.13 − 0.654i)7-s + (0.707 − 0.707i)8-s + (−0.115 + 0.0310i)10-s + (−0.301 − 0.522i)11-s + (0.720 + 0.693i)13-s + (−1.26 − 0.338i)14-s + (0.499 − 0.866i)16-s + (0.783 − 1.35i)17-s + (0.542 − 0.940i)19-s + (−0.103 + 0.0599i)20-s + (−0.426 − 0.426i)22-s + (−0.228 − 0.396i)23-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((0.00641+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00641 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((0.00641+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00641 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 0.00641+0.999i0.00641 + 0.999i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(829,⋅)\chi_{936} (829, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), 0.00641+0.999i)(2,\ 936,\ (\ :1/2),\ 0.00641 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.75456−1.74335i1.75456 - 1.74335i
L(12)L(\frac12) ≈\approx 1.75456−1.74335i1.75456 - 1.74335i
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+(−1.36+0.366i)T 1 + (-1.36 + 0.366i)T
3 1 1
13 1+(−2.59−2.5i)T 1 + (-2.59 - 2.5i)T
good5 1+0.267T+5T2 1 + 0.267T + 5T^{2}
7 1+(3+1.73i)T+(3.5+6.06i)T2 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2}
11 1+(1+1.73i)T+(−5.5+9.52i)T2 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}
17 1+(−3.23+5.59i)T+(−8.5−14.7i)T2 1 + (-3.23 + 5.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.36+4.09i)T+(−9.5−16.4i)T2 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.09+1.90i)T+(−11.5+19.9i)T2 1 + (1.09 + 1.90i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.59+1.5i)T+(14.5−25.1i)T2 1 + (-2.59 + 1.5i)T + (14.5 - 25.1i)T^{2}
31 1+1.26iT−31T2 1 + 1.26iT - 31T^{2}
37 1+(−3.86−6.69i)T+(−18.5+32.0i)T2 1 + (-3.86 - 6.69i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.03+0.598i)T+(20.5−35.5i)T2 1 + (-1.03 + 0.598i)T + (20.5 - 35.5i)T^{2}
43 1+(8.19+4.73i)T+(21.5+37.2i)T2 1 + (8.19 + 4.73i)T + (21.5 + 37.2i)T^{2}
47 1−3.26iT−47T2 1 - 3.26iT - 47T^{2}
53 1−9.92iT−53T2 1 - 9.92iT - 53T^{2}
59 1+(3.73−6.46i)T+(−29.5−51.0i)T2 1 + (3.73 - 6.46i)T + (-29.5 - 51.0i)T^{2}
61 1+(0.866+0.5i)T+(30.5+52.8i)T2 1 + (0.866 + 0.5i)T + (30.5 + 52.8i)T^{2}
67 1+(−5.36−9.29i)T+(−33.5+58.0i)T2 1 + (-5.36 - 9.29i)T + (-33.5 + 58.0i)T^{2}
71 1+(−11.0−6.36i)T+(35.5+61.4i)T2 1 + (-11.0 - 6.36i)T + (35.5 + 61.4i)T^{2}
73 1+1.73iT−73T2 1 + 1.73iT - 73T^{2}
79 1−10.3T+79T2 1 - 10.3T + 79T^{2}
83 1−5.46T+83T2 1 - 5.46T + 83T^{2}
89 1+(−0.464+0.267i)T+(44.5−77.0i)T2 1 + (-0.464 + 0.267i)T + (44.5 - 77.0i)T^{2}
97 1+(−5.19−3i)T+(48.5+84.0i)T2 1 + (-5.19 - 3i)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

−9.927691409183327169359277012311, −9.344406681691370826421050342422, −7.949456959349357993508878631163, −7.00396500620774179454701122414, −6.39460065824097365589405066876, −5.44258615311702612758194650688, −4.37780148477006967609613668145, −3.45630077140752615472690908528, −2.67890572817001675659794805249, −0.861396753172196908325327668250, 1.89887000555132024580175752835, 3.29602147932287314197795783424, 3.74192087355765304226224865282, 5.21106432271597557731762028173, 5.94327190136883027051425537831, 6.51046030466102754969834702749, 7.78126065203882477100397371290, 8.236869317565539712142889930415, 9.612575597763893626074719480585, 10.27187932584092401935472616235

Graph of the ZZ-function along the critical line