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Properties

Label 2-240-5.4-c7-0-14
Degree 22
Conductor 240240
Sign 0.983−0.178i0.983 - 0.178i
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  − 27i·3-s + (−50 − 275i)5-s − 1.12e3i·7-s − 729·9-s + 5.51e3·11-s + 1.27e4i·13-s + (−7.42e3 + 1.35e3i)15-s + 3.22e4i·17-s − 4.44e3·19-s − 3.04e4·21-s + 9.54e4i·23-s + (−7.31e4 + 2.75e4i)25-s + 1.96e4i·27-s − 1.94e4·29-s + 2.40e5·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.178 − 0.983i)5-s − 1.24i·7-s − 0.333·9-s + 1.24·11-s + 1.61i·13-s + (−0.568 + 0.103i)15-s + 1.58i·17-s − 0.148·19-s − 0.716·21-s + 1.63i·23-s + (−0.935 + 0.351i)25-s + 0.192i·27-s − 0.148·29-s + 1.44·31-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=((0.983−0.178i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+7/2) L(s)=((0.983−0.178i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: 0.983−0.178i0.983 - 0.178i
Analytic conductor: 74.972474.9724
Root analytic conductor: 8.658668.65866
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ240(49,⋅)\chi_{240} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 240, ( :7/2), 0.983−0.178i)(2,\ 240,\ (\ :7/2),\ 0.983 - 0.178i)

Particular Values

L(4)L(4) ≈\approx 1.7906120241.790612024
L(12)L(\frac12) ≈\approx 1.7906120241.790612024
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+27iT 1 + 27iT
5 1+(50+275i)T 1 + (50 + 275i)T
good7 1+1.12e3iT−8.23e5T2 1 + 1.12e3iT - 8.23e5T^{2}
11 1−5.51e3T+1.94e7T2 1 - 5.51e3T + 1.94e7T^{2}
13 1−1.27e4iT−6.27e7T2 1 - 1.27e4iT - 6.27e7T^{2}
17 1−3.22e4iT−4.10e8T2 1 - 3.22e4iT - 4.10e8T^{2}
19 1+4.44e3T+8.93e8T2 1 + 4.44e3T + 8.93e8T^{2}
23 1−9.54e4iT−3.40e9T2 1 - 9.54e4iT - 3.40e9T^{2}
29 1+1.94e4T+1.72e10T2 1 + 1.94e4T + 1.72e10T^{2}
31 1−2.40e5T+2.75e10T2 1 - 2.40e5T + 2.75e10T^{2}
37 1+7.78e4iT−9.49e10T2 1 + 7.78e4iT - 9.49e10T^{2}
41 1−2.99e5T+1.94e11T2 1 - 2.99e5T + 1.94e11T^{2}
43 1−4.16e5iT−2.71e11T2 1 - 4.16e5iT - 2.71e11T^{2}
47 1+3.22e5iT−5.06e11T2 1 + 3.22e5iT - 5.06e11T^{2}
53 1−8.80e5iT−1.17e12T2 1 - 8.80e5iT - 1.17e12T^{2}
59 1+1.84e6T+2.48e12T2 1 + 1.84e6T + 2.48e12T^{2}
61 1+8.61e5T+3.14e12T2 1 + 8.61e5T + 3.14e12T^{2}
67 1−6.73e5iT−6.06e12T2 1 - 6.73e5iT - 6.06e12T^{2}
71 1−3.42e6T+9.09e12T2 1 - 3.42e6T + 9.09e12T^{2}
73 1−4.67e6iT−1.10e13T2 1 - 4.67e6iT - 1.10e13T^{2}
79 1+3.13e6T+1.92e13T2 1 + 3.13e6T + 1.92e13T^{2}
83 1−4.84e5iT−2.71e13T2 1 - 4.84e5iT - 2.71e13T^{2}
89 1+6.25e6T+4.42e13T2 1 + 6.25e6T + 4.42e13T^{2}
97 1−8.65e6iT−8.07e13T2 1 - 8.65e6iT - 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

−11.14931798973250606745281236462, −9.766915284900454074061281284781, −8.931319039748286747761811299794, −7.923303658622125574836983466099, −6.92947250545652704520116508396, −6.02989049755201405842633898233, −4.37758051169004290143331145397, −3.84438424475241208460636317415, −1.62500074384728403654676001951, −1.11370687485318844825772264989, 0.47572912326310605033162534962, 2.50985481554923884598977710924, 3.18236816214765662281668435874, 4.62085819448578978027097544718, 5.81282741096414993949558300298, 6.68910779745843748694578937287, 8.012834059151936220076938224074, 9.000419861640643039304646869310, 9.908196050424788283761642943052, 10.83057860108615198882948220536

Graph of the ZZ-function along the critical line