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Properties

Label 2-240-12.11-c7-0-3
Degree 22
Conductor 240240
Sign −0.567+0.823i-0.567 + 0.823i
Analytic cond. 74.972474.9724
Root an. cond. 8.658668.65866
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−42.2 + 20.0i)3-s − 125i·5-s + 1.61e3i·7-s + (1.37e3 − 1.69e3i)9-s − 504.·11-s − 9.49e3·13-s + (2.51e3 + 5.27e3i)15-s − 4.16e3i·17-s + 4.38e4i·19-s + (−3.24e4 − 6.82e4i)21-s + 2.23e4·23-s − 1.56e4·25-s + (−2.41e4 + 9.93e4i)27-s + 2.44e5i·29-s − 5.62e4i·31-s + ⋯
L(s)  = 1  + (−0.902 + 0.429i)3-s − 0.447i·5-s + 1.77i·7-s + (0.630 − 0.775i)9-s − 0.114·11-s − 1.19·13-s + (0.192 + 0.403i)15-s − 0.205i·17-s + 1.46i·19-s + (−0.764 − 1.60i)21-s + 0.383·23-s − 0.199·25-s + (−0.236 + 0.971i)27-s + 1.86i·29-s − 0.339i·31-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=((−0.567+0.823i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+7/2) L(s)=((−0.567+0.823i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: −0.567+0.823i-0.567 + 0.823i
Analytic conductor: 74.972474.9724
Root analytic conductor: 8.658668.65866
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ240(191,⋅)\chi_{240} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 240, ( :7/2), −0.567+0.823i)(2,\ 240,\ (\ :7/2),\ -0.567 + 0.823i)

Particular Values

L(4)L(4) ≈\approx 0.25772222880.2577222288
L(12)L(\frac12) ≈\approx 0.25772222880.2577222288
L(92)L(\frac{9}{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+(42.2−20.0i)T 1 + (42.2 - 20.0i)T
5 1+125iT 1 + 125iT
good7 1−1.61e3iT−8.23e5T2 1 - 1.61e3iT - 8.23e5T^{2}
11 1+504.T+1.94e7T2 1 + 504.T + 1.94e7T^{2}
13 1+9.49e3T+6.27e7T2 1 + 9.49e3T + 6.27e7T^{2}
17 1+4.16e3iT−4.10e8T2 1 + 4.16e3iT - 4.10e8T^{2}
19 1−4.38e4iT−8.93e8T2 1 - 4.38e4iT - 8.93e8T^{2}
23 1−2.23e4T+3.40e9T2 1 - 2.23e4T + 3.40e9T^{2}
29 1−2.44e5iT−1.72e10T2 1 - 2.44e5iT - 1.72e10T^{2}
31 1+5.62e4iT−2.75e10T2 1 + 5.62e4iT - 2.75e10T^{2}
37 1−2.55e5T+9.49e10T2 1 - 2.55e5T + 9.49e10T^{2}
41 1−7.04e4iT−1.94e11T2 1 - 7.04e4iT - 1.94e11T^{2}
43 1−2.40e5iT−2.71e11T2 1 - 2.40e5iT - 2.71e11T^{2}
47 1−5.63e5T+5.06e11T2 1 - 5.63e5T + 5.06e11T^{2}
53 1−1.03e6iT−1.17e12T2 1 - 1.03e6iT - 1.17e12T^{2}
59 1+2.42e6T+2.48e12T2 1 + 2.42e6T + 2.48e12T^{2}
61 1−1.04e6T+3.14e12T2 1 - 1.04e6T + 3.14e12T^{2}
67 1+9.30e4iT−6.06e12T2 1 + 9.30e4iT - 6.06e12T^{2}
71 1+5.15e6T+9.09e12T2 1 + 5.15e6T + 9.09e12T^{2}
73 1+2.53e6T+1.10e13T2 1 + 2.53e6T + 1.10e13T^{2}
79 1+6.21e6iT−1.92e13T2 1 + 6.21e6iT - 1.92e13T^{2}
83 1−9.59e6T+2.71e13T2 1 - 9.59e6T + 2.71e13T^{2}
89 1+9.31e6iT−4.42e13T2 1 + 9.31e6iT - 4.42e13T^{2}
97 1−9.65e5T+8.07e13T2 1 - 9.65e5T + 8.07e13T^{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

−11.74810460562207743279780022929, −10.54233914790877299093251108193, −9.525596316531785793871945175598, −8.865931731999034249789281303483, −7.54720491008424058382829000186, −6.12520230303132060541843856934, −5.42460548646659116617064345039, −4.60413993264008508531882922710, −2.97512214108888258908136564579, −1.57791578142911233161426949434, 0.085652503590857423268690371271, 0.900178725333077761205268349766, 2.45733052016288518910220314788, 4.11526824271605844562389518513, 4.98107155517121817558017798920, 6.42777657222255392719952393062, 7.19597066098583999553780368379, 7.77812078802319043572549845931, 9.627492565564584494931957444507, 10.45728485335317053519255219646

Graph of the ZZ-function along the critical line