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Properties

Label 2-240-12.11-c7-0-49
Degree 22
Conductor 240240
Sign −0.757+0.652i-0.757 + 0.652i
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  + (15.4 − 44.1i)3-s − 125i·5-s − 3.33i·7-s + (−1.71e3 − 1.36e3i)9-s + 833.·11-s + 1.22e4·13-s + (−5.51e3 − 1.92e3i)15-s + 1.20e4i·17-s − 3.83e4i·19-s + (−147. − 51.4i)21-s + 1.07e5·23-s − 1.56e4·25-s + (−8.65e4 + 5.44e4i)27-s − 2.02e5i·29-s − 3.65e4i·31-s + ⋯
L(s)  = 1  + (0.330 − 0.943i)3-s − 0.447i·5-s − 0.00367i·7-s + (−0.782 − 0.623i)9-s + 0.188·11-s + 1.54·13-s + (−0.422 − 0.147i)15-s + 0.592i·17-s − 1.28i·19-s + (−0.00346 − 0.00121i)21-s + 1.84·23-s − 0.199·25-s + (−0.846 + 0.532i)27-s − 1.54i·29-s − 0.220i·31-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=((−0.757+0.652i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+7/2) L(s)=((−0.757+0.652i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: −0.757+0.652i-0.757 + 0.652i
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), −0.757+0.652i)(2,\ 240,\ (\ :7/2),\ -0.757 + 0.652i)

Particular Values

L(4)L(4) ≈\approx 2.3262738902.326273890
L(12)L(\frac12) ≈\approx 2.3262738902.326273890
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+(−15.4+44.1i)T 1 + (-15.4 + 44.1i)T
5 1+125iT 1 + 125iT
good7 1+3.33iT−8.23e5T2 1 + 3.33iT - 8.23e5T^{2}
11 1−833.T+1.94e7T2 1 - 833.T + 1.94e7T^{2}
13 1−1.22e4T+6.27e7T2 1 - 1.22e4T + 6.27e7T^{2}
17 1−1.20e4iT−4.10e8T2 1 - 1.20e4iT - 4.10e8T^{2}
19 1+3.83e4iT−8.93e8T2 1 + 3.83e4iT - 8.93e8T^{2}
23 1−1.07e5T+3.40e9T2 1 - 1.07e5T + 3.40e9T^{2}
29 1+2.02e5iT−1.72e10T2 1 + 2.02e5iT - 1.72e10T^{2}
31 1+3.65e4iT−2.75e10T2 1 + 3.65e4iT - 2.75e10T^{2}
37 1+3.38e5T+9.49e10T2 1 + 3.38e5T + 9.49e10T^{2}
41 1+3.13e5iT−1.94e11T2 1 + 3.13e5iT - 1.94e11T^{2}
43 1+6.19e4iT−2.71e11T2 1 + 6.19e4iT - 2.71e11T^{2}
47 1−9.58e5T+5.06e11T2 1 - 9.58e5T + 5.06e11T^{2}
53 1−2.57e5iT−1.17e12T2 1 - 2.57e5iT - 1.17e12T^{2}
59 1+1.95e6T+2.48e12T2 1 + 1.95e6T + 2.48e12T^{2}
61 1+1.08e6T+3.14e12T2 1 + 1.08e6T + 3.14e12T^{2}
67 1−8.33e5iT−6.06e12T2 1 - 8.33e5iT - 6.06e12T^{2}
71 1+2.51e6T+9.09e12T2 1 + 2.51e6T + 9.09e12T^{2}
73 1+2.38e6T+1.10e13T2 1 + 2.38e6T + 1.10e13T^{2}
79 1−4.63e6iT−1.92e13T2 1 - 4.63e6iT - 1.92e13T^{2}
83 1+1.20e6T+2.71e13T2 1 + 1.20e6T + 2.71e13T^{2}
89 1+3.27e6iT−4.42e13T2 1 + 3.27e6iT - 4.42e13T^{2}
97 1−1.27e7T+8.07e13T2 1 - 1.27e7T + 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

−10.71427033685788067353754592978, −9.044064026343321003920515265233, −8.712806248453018176396693516113, −7.51559944554271277049552910963, −6.54378513155396122767861945546, −5.57925104285012608957300313987, −4.06485814891392858466579122261, −2.80703069304025603821589794600, −1.46070844261841757809377837132, −0.56137171839748330568801238138, 1.34301237768354099566999306748, 3.00651841134182930881314840640, 3.73231160853729408418532894945, 5.02322351022455597385254714139, 6.09751704092242177844137859060, 7.34705249827500725934720844480, 8.619562568764268986068576919528, 9.195785715667921476670907343591, 10.51667015433800246233103177817, 10.87653922913998762757779473541

Graph of the ZZ-function along the critical line