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Properties

Label 2-240-12.11-c7-0-7
Degree 22
Conductor 240240
Sign 0.186−0.982i0.186 - 0.982i
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)=((0.186−0.982i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+7/2) L(s)=((0.186−0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: 0.186−0.982i0.186 - 0.982i
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), 0.186−0.982i)(2,\ 240,\ (\ :7/2),\ 0.186 - 0.982i)

Particular Values

L(4)L(4) ≈\approx 0.82687488180.8268748818
L(12)L(\frac12) ≈\approx 0.82687488180.8268748818
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

−11.21249895730605933746388869992, −10.38502527133379795050820541609, −8.944545590624697349171498080904, −8.056073751512720466592788160265, −6.97953821321695077082210885762, −6.24157195655739024356887408091, −5.15163830592915466837612061349, −3.53418035884994182907204649710, −2.27958748893752892443783836821, −1.06048637444739748311345784526, 0.22459794957348760514332704490, 1.73813064443906384602850779939, 3.57608390618044859100728184057, 4.22313480415061844759790109959, 5.65554497043317370420790592134, 6.19594867936565473254659271645, 8.050760303146312798480463000711, 8.687303146423792273698868734844, 9.900468515481859755572617096895, 10.50575256440786627547740773015

Graph of the ZZ-function along the critical line