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Properties

Label 2-3240-360.259-c0-0-9
Degree 22
Conductor 32403240
Sign 0.984+0.173i0.984 + 0.173i
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)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.866 − 1.5i)7-s + 0.999·8-s − 0.999·10-s + (−0.866 − 1.5i)13-s + (0.866 + 1.5i)14-s + (−0.5 + 0.866i)16-s + 19-s + (0.499 − 0.866i)20-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + 1.73·26-s − 1.73·28-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.866 − 1.5i)7-s + 0.999·8-s − 0.999·10-s + (−0.866 − 1.5i)13-s + (0.866 + 1.5i)14-s + (−0.5 + 0.866i)16-s + 19-s + (0.499 − 0.866i)20-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + 1.73·26-s − 1.73·28-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.984+0.173i0.984 + 0.173i
Analytic conductor: 1.616971.61697
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3240(379,⋅)\chi_{3240} (379, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :0), 0.984+0.173i)(2,\ 3240,\ (\ :0),\ 0.984 + 0.173i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0290136561.029013656
L(12)L(\frac12) ≈\approx 1.0290136561.029013656
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1 1
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good7 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
11 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
13 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1−T+T2 1 - T + T^{2}
23 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
29 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+T2 1 + T^{2}
41 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+T+T2 1 + T + T^{2}
59 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)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

−8.571296513410206845909214755927, −7.86366843866841084614309041802, −7.27516490243461534720762930013, −6.93448063066746972123759387568, −5.80010589999963555904474513588, −5.22118609886935278902269246630, −4.34904785810865522909886921583, −3.31666941922939719615505666243, −2.03615670931660941890564309877, −0.76791504493148578925639947913, 1.53383108504825593926054481878, 2.00989816245952511287976353807, 2.97085189344275941098175467200, 4.29290507556378104956988255263, 4.95314746965196723847036818141, 5.52958942153002377649906685039, 6.65276122278699662168649065487, 7.80609273040622150595082547335, 8.250883407629305071405093070438, 9.184579715381157226212144828112

Graph of the ZZ-function along the critical line