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Properties

Label 2-3240-9.7-c1-0-17
Degree 22
Conductor 32403240
Sign 0.173−0.984i0.173 - 0.984i
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
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  + (0.5 + 0.866i)5-s + (−1.68 + 2.92i)7-s + (1.18 − 2.05i)11-s + (1.68 + 2.92i)13-s + 6.74·17-s + 19-s + (2.68 + 4.65i)23-s + (−0.499 + 0.866i)25-s + (1.18 − 2.05i)29-s + (−5.55 − 9.62i)31-s − 3.37·35-s + 6·37-s + (0.127 + 0.221i)41-s + (2.37 − 4.10i)43-s + (−4.68 + 8.11i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.637 + 1.10i)7-s + (0.357 − 0.619i)11-s + (0.467 + 0.809i)13-s + 1.63·17-s + 0.229·19-s + (0.560 + 0.970i)23-s + (−0.0999 + 0.173i)25-s + (0.220 − 0.381i)29-s + (−0.998 − 1.72i)31-s − 0.570·35-s + 0.986·37-s + (0.0199 + 0.0345i)41-s + (0.361 − 0.626i)43-s + (−0.683 + 1.18i)47-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.173−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=((0.173−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.173−0.984i0.173 - 0.984i
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3240(2161,⋅)\chi_{3240} (2161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :1/2), 0.173−0.984i)(2,\ 3240,\ (\ :1/2),\ 0.173 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 1.8752373161.875237316
L(12)L(\frac12) ≈\approx 1.8752373161.875237316
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
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good7 1+(1.68−2.92i)T+(−3.5−6.06i)T2 1 + (1.68 - 2.92i)T + (-3.5 - 6.06i)T^{2}
11 1+(−1.18+2.05i)T+(−5.5−9.52i)T2 1 + (-1.18 + 2.05i)T + (-5.5 - 9.52i)T^{2}
13 1+(−1.68−2.92i)T+(−6.5+11.2i)T2 1 + (-1.68 - 2.92i)T + (-6.5 + 11.2i)T^{2}
17 1−6.74T+17T2 1 - 6.74T + 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1+(−2.68−4.65i)T+(−11.5+19.9i)T2 1 + (-2.68 - 4.65i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.18+2.05i)T+(−14.5−25.1i)T2 1 + (-1.18 + 2.05i)T + (-14.5 - 25.1i)T^{2}
31 1+(5.55+9.62i)T+(−15.5+26.8i)T2 1 + (5.55 + 9.62i)T + (-15.5 + 26.8i)T^{2}
37 1−6T+37T2 1 - 6T + 37T^{2}
41 1+(−0.127−0.221i)T+(−20.5+35.5i)T2 1 + (-0.127 - 0.221i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2.37+4.10i)T+(−21.5−37.2i)T2 1 + (-2.37 + 4.10i)T + (-21.5 - 37.2i)T^{2}
47 1+(4.68−8.11i)T+(−23.5−40.7i)T2 1 + (4.68 - 8.11i)T + (-23.5 - 40.7i)T^{2}
53 1+10.1T+53T2 1 + 10.1T + 53T^{2}
59 1+(−2.5−4.33i)T+(−29.5+51.0i)T2 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.37−11.0i)T+(−30.5−52.8i)T2 1 + (6.37 - 11.0i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.372−0.644i)T+(−33.5+58.0i)T2 1 + (-0.372 - 0.644i)T + (-33.5 + 58.0i)T^{2}
71 1−4.37T+71T2 1 - 4.37T + 71T^{2}
73 1−14.7T+73T2 1 - 14.7T + 73T^{2}
79 1+(−1.37+2.37i)T+(−39.5−68.4i)T2 1 + (-1.37 + 2.37i)T + (-39.5 - 68.4i)T^{2}
83 1+(5−8.66i)T+(−41.5−71.8i)T2 1 + (5 - 8.66i)T + (-41.5 - 71.8i)T^{2}
89 1−4.37T+89T2 1 - 4.37T + 89T^{2}
97 1+(−2.37+4.10i)T+(−48.5−84.0i)T2 1 + (-2.37 + 4.10i)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

−9.112783802350388738412434397114, −8.004662167093226283376283803339, −7.41014024188427632536959824248, −6.26406453239641901429180780728, −5.98208856926853479526461508057, −5.23706705582916443126165243521, −3.93021461278635452995412584924, −3.23919982772485728603187505649, −2.39379911977339480225420684982, −1.18736198595815857656106018395, 0.66615072709403905147191067825, 1.55121773493558672883958471402, 3.11879955324003492350427039399, 3.60148448259644478562198771181, 4.69608712032358468060580860442, 5.35588359540097145314319695253, 6.35099772989000685213767813321, 6.96814818203086534525728437085, 7.74736746467743313249571357043, 8.400077895531641708352019145881

Graph of the ZZ-function along the critical line