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Properties

Label 2-936-9.7-c1-0-10
Degree 22
Conductor 936936
Sign 0.722−0.691i0.722 - 0.691i
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  + (−1.69 + 0.356i)3-s + (0.348 + 0.603i)5-s + (−0.902 + 1.56i)7-s + (2.74 − 1.20i)9-s + (1.77 − 3.07i)11-s + (−0.5 − 0.866i)13-s + (−0.805 − 0.898i)15-s + 4.15·17-s − 3.92·19-s + (0.973 − 2.97i)21-s + (1.22 + 2.12i)23-s + (2.25 − 3.91i)25-s + (−4.22 + 3.02i)27-s + (−4.05 + 7.02i)29-s + (2.87 + 4.98i)31-s + ⋯
L(s)  = 1  + (−0.978 + 0.205i)3-s + (0.155 + 0.269i)5-s + (−0.341 + 0.590i)7-s + (0.915 − 0.402i)9-s + (0.535 − 0.927i)11-s + (−0.138 − 0.240i)13-s + (−0.207 − 0.231i)15-s + 1.00·17-s − 0.900·19-s + (0.212 − 0.648i)21-s + (0.255 + 0.442i)23-s + (0.451 − 0.782i)25-s + (−0.813 + 0.582i)27-s + (−0.752 + 1.30i)29-s + (0.517 + 0.895i)31-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((0.722−0.691i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((0.722−0.691i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 0.722−0.691i0.722 - 0.691i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(313,⋅)\chi_{936} (313, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), 0.722−0.691i)(2,\ 936,\ (\ :1/2),\ 0.722 - 0.691i)

Particular Values

L(1)L(1) ≈\approx 1.04822+0.420668i1.04822 + 0.420668i
L(12)L(\frac12) ≈\approx 1.04822+0.420668i1.04822 + 0.420668i
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
3 1+(1.69−0.356i)T 1 + (1.69 - 0.356i)T
13 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good5 1+(−0.348−0.603i)T+(−2.5+4.33i)T2 1 + (-0.348 - 0.603i)T + (-2.5 + 4.33i)T^{2}
7 1+(0.902−1.56i)T+(−3.5−6.06i)T2 1 + (0.902 - 1.56i)T + (-3.5 - 6.06i)T^{2}
11 1+(−1.77+3.07i)T+(−5.5−9.52i)T2 1 + (-1.77 + 3.07i)T + (-5.5 - 9.52i)T^{2}
17 1−4.15T+17T2 1 - 4.15T + 17T^{2}
19 1+3.92T+19T2 1 + 3.92T + 19T^{2}
23 1+(−1.22−2.12i)T+(−11.5+19.9i)T2 1 + (-1.22 - 2.12i)T + (-11.5 + 19.9i)T^{2}
29 1+(4.05−7.02i)T+(−14.5−25.1i)T2 1 + (4.05 - 7.02i)T + (-14.5 - 25.1i)T^{2}
31 1+(−2.87−4.98i)T+(−15.5+26.8i)T2 1 + (-2.87 - 4.98i)T + (-15.5 + 26.8i)T^{2}
37 1−11.3T+37T2 1 - 11.3T + 37T^{2}
41 1+(−2.04−3.54i)T+(−20.5+35.5i)T2 1 + (-2.04 - 3.54i)T + (-20.5 + 35.5i)T^{2}
43 1+(−4.75+8.22i)T+(−21.5−37.2i)T2 1 + (-4.75 + 8.22i)T + (-21.5 - 37.2i)T^{2}
47 1+(6.09−10.5i)T+(−23.5−40.7i)T2 1 + (6.09 - 10.5i)T + (-23.5 - 40.7i)T^{2}
53 1−5.68T+53T2 1 - 5.68T + 53T^{2}
59 1+(0.315+0.546i)T+(−29.5+51.0i)T2 1 + (0.315 + 0.546i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.556+0.963i)T+(−30.5−52.8i)T2 1 + (-0.556 + 0.963i)T + (-30.5 - 52.8i)T^{2}
67 1+(−7.05−12.2i)T+(−33.5+58.0i)T2 1 + (-7.05 - 12.2i)T + (-33.5 + 58.0i)T^{2}
71 1−4.85T+71T2 1 - 4.85T + 71T^{2}
73 1+0.187T+73T2 1 + 0.187T + 73T^{2}
79 1+(−1.82+3.16i)T+(−39.5−68.4i)T2 1 + (-1.82 + 3.16i)T + (-39.5 - 68.4i)T^{2}
83 1+(−1.34+2.33i)T+(−41.5−71.8i)T2 1 + (-1.34 + 2.33i)T + (-41.5 - 71.8i)T^{2}
89 1+7.04T+89T2 1 + 7.04T + 89T^{2}
97 1+(−6.95+12.0i)T+(−48.5−84.0i)T2 1 + (-6.95 + 12.0i)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

−10.27863838984935229718631217455, −9.430484251833167856007757804409, −8.635972928885454096783465340893, −7.49326303619932994454338050514, −6.44857832789033632043362846342, −5.93932989782143241448775666735, −5.08408856476994138792122808660, −3.90239534774249854337694150691, −2.81662922118837029766478487300, −1.06565837781319769034503606390, 0.78438853741718652349538564293, 2.13928582058603408450679116973, 3.93599019136112725440517902247, 4.62117949801900354330578885540, 5.70876394603759725952761055041, 6.51885605950446669672120373834, 7.26752562125098708601498896212, 8.081810373637495915193395314422, 9.551242282592104987436809337506, 9.824234844208505407658329357398

Graph of the ZZ-function along the critical line