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Properties

Label 2-240-12.11-c7-0-19
Degree 22
Conductor 240240
Sign 0.0422−0.999i0.0422 - 0.999i
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  + (46.7 + 1.97i)3-s + 125i·5-s − 710. i·7-s + (2.17e3 + 184. i)9-s + 2.11e3·11-s − 1.38e4·13-s + (−246. + 5.84e3i)15-s + 2.39e4i·17-s + 2.80e4i·19-s + (1.40e3 − 3.32e4i)21-s + 4.68e3·23-s − 1.56e4·25-s + (1.01e5 + 1.29e4i)27-s + 2.04e5i·29-s − 2.93e5i·31-s + ⋯
L(s)  = 1  + (0.999 + 0.0422i)3-s + 0.447i·5-s − 0.783i·7-s + (0.996 + 0.0843i)9-s + 0.478·11-s − 1.75·13-s + (−0.0188 + 0.446i)15-s + 1.18i·17-s + 0.937i·19-s + (0.0330 − 0.782i)21-s + 0.0803·23-s − 0.199·25-s + (0.991 + 0.126i)27-s + 1.55i·29-s − 1.76i·31-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=((0.0422−0.999i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0422 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+7/2) L(s)=((0.0422−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0422 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: 0.0422−0.999i0.0422 - 0.999i
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), 0.0422−0.999i)(2,\ 240,\ (\ :7/2),\ 0.0422 - 0.999i)

Particular Values

L(4)L(4) ≈\approx 2.4754599512.475459951
L(12)L(\frac12) ≈\approx 2.4754599512.475459951
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+(−46.7−1.97i)T 1 + (-46.7 - 1.97i)T
5 1−125iT 1 - 125iT
good7 1+710.iT−8.23e5T2 1 + 710. iT - 8.23e5T^{2}
11 1−2.11e3T+1.94e7T2 1 - 2.11e3T + 1.94e7T^{2}
13 1+1.38e4T+6.27e7T2 1 + 1.38e4T + 6.27e7T^{2}
17 1−2.39e4iT−4.10e8T2 1 - 2.39e4iT - 4.10e8T^{2}
19 1−2.80e4iT−8.93e8T2 1 - 2.80e4iT - 8.93e8T^{2}
23 1−4.68e3T+3.40e9T2 1 - 4.68e3T + 3.40e9T^{2}
29 1−2.04e5iT−1.72e10T2 1 - 2.04e5iT - 1.72e10T^{2}
31 1+2.93e5iT−2.75e10T2 1 + 2.93e5iT - 2.75e10T^{2}
37 1−1.33e5T+9.49e10T2 1 - 1.33e5T + 9.49e10T^{2}
41 1−6.93e5iT−1.94e11T2 1 - 6.93e5iT - 1.94e11T^{2}
43 1−8.24e5iT−2.71e11T2 1 - 8.24e5iT - 2.71e11T^{2}
47 1−1.08e6T+5.06e11T2 1 - 1.08e6T + 5.06e11T^{2}
53 1+2.96e5iT−1.17e12T2 1 + 2.96e5iT - 1.17e12T^{2}
59 1+2.40e6T+2.48e12T2 1 + 2.40e6T + 2.48e12T^{2}
61 1+1.95e6T+3.14e12T2 1 + 1.95e6T + 3.14e12T^{2}
67 1+2.22e5iT−6.06e12T2 1 + 2.22e5iT - 6.06e12T^{2}
71 1−1.63e6T+9.09e12T2 1 - 1.63e6T + 9.09e12T^{2}
73 1+4.96e4T+1.10e13T2 1 + 4.96e4T + 1.10e13T^{2}
79 1−6.73e6iT−1.92e13T2 1 - 6.73e6iT - 1.92e13T^{2}
83 1−6.45e6T+2.71e13T2 1 - 6.45e6T + 2.71e13T^{2}
89 1+5.16e6iT−4.42e13T2 1 + 5.16e6iT - 4.42e13T^{2}
97 1+3.73e5T+8.07e13T2 1 + 3.73e5T + 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

−10.86583961246154842286015045128, −10.01551562241565250527643701013, −9.326868050498195763371628094765, −7.955144657258949205936519928085, −7.41392526688764706306006315429, −6.29124476299503315317838534783, −4.59275113837848545194184060924, −3.68633525157998530095037753849, −2.52620071081362118251234267322, −1.32630544835107594578023579005, 0.49257028280582247428447425915, 2.10502995843730768355131255957, 2.86449643246279160492718948842, 4.38276116977283812426064538822, 5.30050343878130387510897865629, 6.92661483237151027029569815301, 7.67931746579278852665426225122, 9.066192318359656985389630765166, 9.207665972381308449238227503482, 10.41013870251154974245769881294

Graph of the ZZ-function along the critical line