↑

Properties

Label 2-240-12.11-c7-0-46
Degree 22
Conductor 240240
Sign −0.317+0.948i-0.317 + 0.948i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (35.0 − 30.9i)3-s + 125i·5-s + 201. i·7-s + (267. − 2.17e3i)9-s − 499.·11-s + 1.35e3·13-s + (3.87e3 + 4.37e3i)15-s − 1.67e3i·17-s − 1.61e4i·19-s + (6.23e3 + 7.05e3i)21-s − 1.68e4·23-s − 1.56e4·25-s + (−5.78e4 − 8.43e4i)27-s + 8.26e4i·29-s − 1.61e5i·31-s + ⋯
L(s)  = 1  + (0.749 − 0.662i)3-s + 0.447i·5-s + 0.221i·7-s + (0.122 − 0.992i)9-s − 0.113·11-s + 0.171·13-s + (0.296 + 0.335i)15-s − 0.0824i·17-s − 0.539i·19-s + (0.146 + 0.166i)21-s − 0.288·23-s − 0.199·25-s + (−0.565 − 0.824i)27-s + 0.628i·29-s − 0.975i·31-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=((−0.317+0.948i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+7/2) L(s)=((−0.317+0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: −0.317+0.948i-0.317 + 0.948i
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), −0.317+0.948i)(2,\ 240,\ (\ :7/2),\ -0.317 + 0.948i)

Particular Values

L(4)L(4) ≈\approx 2.1821607622.182160762
L(12)L(\frac12) ≈\approx 2.1821607622.182160762
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+(−35.0+30.9i)T 1 + (-35.0 + 30.9i)T
5 1−125iT 1 - 125iT
good7 1−201.iT−8.23e5T2 1 - 201. iT - 8.23e5T^{2}
11 1+499.T+1.94e7T2 1 + 499.T + 1.94e7T^{2}
13 1−1.35e3T+6.27e7T2 1 - 1.35e3T + 6.27e7T^{2}
17 1+1.67e3iT−4.10e8T2 1 + 1.67e3iT - 4.10e8T^{2}
19 1+1.61e4iT−8.93e8T2 1 + 1.61e4iT - 8.93e8T^{2}
23 1+1.68e4T+3.40e9T2 1 + 1.68e4T + 3.40e9T^{2}
29 1−8.26e4iT−1.72e10T2 1 - 8.26e4iT - 1.72e10T^{2}
31 1+1.61e5iT−2.75e10T2 1 + 1.61e5iT - 2.75e10T^{2}
37 1−1.28e5T+9.49e10T2 1 - 1.28e5T + 9.49e10T^{2}
41 1+2.74e5iT−1.94e11T2 1 + 2.74e5iT - 1.94e11T^{2}
43 1+7.66e5iT−2.71e11T2 1 + 7.66e5iT - 2.71e11T^{2}
47 1−5.58e5T+5.06e11T2 1 - 5.58e5T + 5.06e11T^{2}
53 1+1.13e6iT−1.17e12T2 1 + 1.13e6iT - 1.17e12T^{2}
59 1+8.80e5T+2.48e12T2 1 + 8.80e5T + 2.48e12T^{2}
61 1+7.56e5T+3.14e12T2 1 + 7.56e5T + 3.14e12T^{2}
67 1+1.60e6iT−6.06e12T2 1 + 1.60e6iT - 6.06e12T^{2}
71 1−1.92e6T+9.09e12T2 1 - 1.92e6T + 9.09e12T^{2}
73 1+4.17e6T+1.10e13T2 1 + 4.17e6T + 1.10e13T^{2}
79 1−1.22e6iT−1.92e13T2 1 - 1.22e6iT - 1.92e13T^{2}
83 1−1.39e6T+2.71e13T2 1 - 1.39e6T + 2.71e13T^{2}
89 1+3.73e6iT−4.42e13T2 1 + 3.73e6iT - 4.42e13T^{2}
97 1+1.25e7T+8.07e13T2 1 + 1.25e7T + 8.07e13T^{2}
show more
show less
   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

−10.56963881311874958432013967061, −9.439586976093549337596965859836, −8.601185842616798064446427748045, −7.57631741490184507205491738911, −6.76406798340059992355984645633, −5.65274490385765341671781317863, −4.02763295782811136151955882526, −2.88339855313704585410182278525, −1.90898198435224309928038010046, −0.45935378467578438783206655509, 1.29069990088765852215503048955, 2.66039421197466712649565726743, 3.86886562257035419539623235744, 4.76527213165705078194720953847, 5.98847455665653116798337495584, 7.50379197495358909229073653190, 8.326066872531213868968729094637, 9.233956970089979250486431149076, 10.09800345270737123670532323966, 10.95178400766299558999932061600

Graph of the ZZ-function along the critical line