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Properties

Label 2-240-12.11-c7-0-26
Degree 22
Conductor 240240
Sign 0.115+0.993i0.115 + 0.993i
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  + (−27.9 − 37.5i)3-s − 125i·5-s + 1.23e3i·7-s + (−628. + 2.09e3i)9-s − 4.91e3·11-s − 7.45e3·13-s + (−4.68e3 + 3.48e3i)15-s + 1.92e4i·17-s − 2.59e4i·19-s + (4.62e4 − 3.44e4i)21-s − 3.33e4·23-s − 1.56e4·25-s + (9.61e4 − 3.49e4i)27-s + 8.13e3i·29-s + 2.00e5i·31-s + ⋯
L(s)  = 1  + (−0.596 − 0.802i)3-s − 0.447i·5-s + 1.35i·7-s + (−0.287 + 0.957i)9-s − 1.11·11-s − 0.941·13-s + (−0.358 + 0.266i)15-s + 0.949i·17-s − 0.867i·19-s + (1.09 − 0.811i)21-s − 0.572·23-s − 0.199·25-s + (0.939 − 0.341i)27-s + 0.0619i·29-s + 1.20i·31-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=((0.115+0.993i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+7/2) L(s)=((0.115+0.993i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: 0.115+0.993i0.115 + 0.993i
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), 0.115+0.993i)(2,\ 240,\ (\ :7/2),\ 0.115 + 0.993i)

Particular Values

L(4)L(4) ≈\approx 0.76870609700.7687060970
L(12)L(\frac12) ≈\approx 0.76870609700.7687060970
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+(27.9+37.5i)T 1 + (27.9 + 37.5i)T
5 1+125iT 1 + 125iT
good7 1−1.23e3iT−8.23e5T2 1 - 1.23e3iT - 8.23e5T^{2}
11 1+4.91e3T+1.94e7T2 1 + 4.91e3T + 1.94e7T^{2}
13 1+7.45e3T+6.27e7T2 1 + 7.45e3T + 6.27e7T^{2}
17 1−1.92e4iT−4.10e8T2 1 - 1.92e4iT - 4.10e8T^{2}
19 1+2.59e4iT−8.93e8T2 1 + 2.59e4iT - 8.93e8T^{2}
23 1+3.33e4T+3.40e9T2 1 + 3.33e4T + 3.40e9T^{2}
29 1−8.13e3iT−1.72e10T2 1 - 8.13e3iT - 1.72e10T^{2}
31 1−2.00e5iT−2.75e10T2 1 - 2.00e5iT - 2.75e10T^{2}
37 1−4.91e5T+9.49e10T2 1 - 4.91e5T + 9.49e10T^{2}
41 1+7.75e5iT−1.94e11T2 1 + 7.75e5iT - 1.94e11T^{2}
43 1−9.88e5iT−2.71e11T2 1 - 9.88e5iT - 2.71e11T^{2}
47 1+5.90e5T+5.06e11T2 1 + 5.90e5T + 5.06e11T^{2}
53 1+1.50e6iT−1.17e12T2 1 + 1.50e6iT - 1.17e12T^{2}
59 1+1.08e6T+2.48e12T2 1 + 1.08e6T + 2.48e12T^{2}
61 1−1.41e6T+3.14e12T2 1 - 1.41e6T + 3.14e12T^{2}
67 1+4.00e6iT−6.06e12T2 1 + 4.00e6iT - 6.06e12T^{2}
71 1−2.32e6T+9.09e12T2 1 - 2.32e6T + 9.09e12T^{2}
73 1−3.70e6T+1.10e13T2 1 - 3.70e6T + 1.10e13T^{2}
79 1+1.74e3iT−1.92e13T2 1 + 1.74e3iT - 1.92e13T^{2}
83 1+7.56e6T+2.71e13T2 1 + 7.56e6T + 2.71e13T^{2}
89 1−2.08e6iT−4.42e13T2 1 - 2.08e6iT - 4.42e13T^{2}
97 1+1.45e5T+8.07e13T2 1 + 1.45e5T + 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

−10.84744247238505965595465050729, −9.670288349909910340445934148953, −8.489677012033101993056008635811, −7.78179919868960631990100512159, −6.50638789200227374603559476290, −5.51060871828514936647818216103, −4.84026782981880964226633410968, −2.72614506018246209715855224685, −1.87210814594785999404838488680, −0.29718273090387220190604449493, 0.67567222421103645059693274752, 2.64859517829863484289212316043, 3.91731451564268857038310194081, 4.81451507219215283641177842670, 5.94067566972625134237699127268, 7.16842333407367723538036943247, 7.944994101440147221188236685025, 9.712878158420956583767518271621, 10.08223323535802649606618813117, 10.96235504353082041528754378377

Graph of the ZZ-function along the critical line