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Properties

Label 2-3240-45.22-c0-0-0
Degree 22
Conductor 32403240
Sign 0.0572−0.998i0.0572 - 0.998i
Analytic cond. 1.616971.61697
Root an. cond. 1.271601.27160
Motivic weight 00
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 + (0.5 + 0.866i)11-s − i·19-s + (0.366 + 1.36i)23-s + (−0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.5 + 0.866i)31-s + (1 + i)37-s + (−0.5 + 0.866i)41-s + (−1.36 − 0.366i)43-s + (−0.366 + 1.36i)47-s + (0.866 + 0.5i)49-s − 0.999·55-s + (−0.866 − 0.5i)59-s + (−1.36 + 0.366i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)11-s − i·19-s + (0.366 + 1.36i)23-s + (−0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.5 + 0.866i)31-s + (1 + i)37-s + (−0.5 + 0.866i)41-s + (−1.36 − 0.366i)43-s + (−0.366 + 1.36i)47-s + (0.866 + 0.5i)49-s − 0.999·55-s + (−0.866 − 0.5i)59-s + (−1.36 + 0.366i)67-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.0572−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0572 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s) L(s)=((0.0572−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0572 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.0572−0.998i0.0572 - 0.998i
Analytic conductor: 1.616971.61697
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3240(2377,⋅)\chi_{3240} (2377, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :0), 0.0572−0.998i)(2,\ 3240,\ (\ :0),\ 0.0572 - 0.998i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0537366691.053736669
L(12)L(\frac12) ≈\approx 1.0537366691.053736669
L(1)L(1) 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+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
17 1+iT2 1 + iT^{2}
19 1+iT−T2 1 + iT - T^{2}
23 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
29 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
31 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
41 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(1.36+0.366i)T+(0.866+0.5i)T2 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2}
47 1+(0.366−1.36i)T+(−0.866−0.5i)T2 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2}
53 1−iT2 1 - iT^{2}
59 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
61 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
67 1+(1.36−0.366i)T+(0.866−0.5i)T2 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2}
71 1+T+T2 1 + T + T^{2}
73 1−iT2 1 - iT^{2}
79 1+(−1.73+i)T+(0.5−0.866i)T2 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}
83 1+(−1.36−0.366i)T+(0.866+0.5i)T2 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2}
89 1−iT−T2 1 - iT - T^{2}
97 1+(0.366−1.36i)T+(−0.866−0.5i)T2 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)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

−9.154473359471798235938521563944, −8.081130574678612484007986221795, −7.51428295178064479261775509314, −6.75305110262426477727581411544, −6.28876774820868120322483359642, −5.04086150914013921256310793949, −4.39966533212834919477159111666, −3.39336745468154481034985422019, −2.69975141606268938637675253478, −1.45498677274894673639016116156, 0.67546557150675981203024639222, 1.89575732645684157178202346055, 3.22670852639926538921585103035, 3.98189691052450903541596376363, 4.75008053349567689454058351269, 5.62890187839233156463048863031, 6.32112476942855722632999851409, 7.24936844627406609799534562023, 8.077962869407071539825627935258, 8.658373309967861210365298090010

Graph of the ZZ-function along the critical line