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Properties

Label 2-1530-5.4-c1-0-11
Degree 22
Conductor 15301530
Sign 0.970+0.241i0.970 + 0.241i
Analytic cond. 12.217112.2171
Root an. cond. 3.495293.49529
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + (−2.17 − 0.539i)5-s − 1.07i·7-s + i·8-s + (−0.539 + 2.17i)10-s + 4·11-s + 5.41i·13-s − 1.07·14-s + 16-s − i·17-s − 8.68·19-s + (2.17 + 0.539i)20-s − 4i·22-s + 0.921i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.970 − 0.241i)5-s − 0.407i·7-s + 0.353i·8-s + (−0.170 + 0.686i)10-s + 1.20·11-s + 1.50i·13-s − 0.288·14-s + 0.250·16-s − 0.242i·17-s − 1.99·19-s + (0.485 + 0.120i)20-s − 0.852i·22-s + 0.192i·23-s + ⋯

Functional equation

Λ(s)=(1530s/2 ΓC(s) L(s)=((0.970+0.241i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1530s/2 ΓC(s+1/2) L(s)=((0.970+0.241i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15301530    =    2⋅32⋅5⋅172 \cdot 3^{2} \cdot 5 \cdot 17
Sign: 0.970+0.241i0.970 + 0.241i
Analytic conductor: 12.217112.2171
Root analytic conductor: 3.495293.49529
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1530(919,⋅)\chi_{1530} (919, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1530, ( :1/2), 0.970+0.241i)(2,\ 1530,\ (\ :1/2),\ 0.970 + 0.241i)

Particular Values

L(1)L(1) ≈\approx 1.2027908641.202790864
L(12)L(\frac12) ≈\approx 1.2027908641.202790864
L(32)L(\frac{3}{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+iT 1 + iT
3 1 1
5 1+(2.17+0.539i)T 1 + (2.17 + 0.539i)T
17 1+iT 1 + iT
good7 1+1.07iT−7T2 1 + 1.07iT - 7T^{2}
11 1−4T+11T2 1 - 4T + 11T^{2}
13 1−5.41iT−13T2 1 - 5.41iT - 13T^{2}
19 1+8.68T+19T2 1 + 8.68T + 19T^{2}
23 1−0.921iT−23T2 1 - 0.921iT - 23T^{2}
29 1−4.34T+29T2 1 - 4.34T + 29T^{2}
31 1−3.07T+31T2 1 - 3.07T + 31T^{2}
37 1−8.34iT−37T2 1 - 8.34iT - 37T^{2}
41 1−3.26T+41T2 1 - 3.26T + 41T^{2}
43 1−9.41iT−43T2 1 - 9.41iT - 43T^{2}
47 1+2.15iT−47T2 1 + 2.15iT - 47T^{2}
53 1+13.5iT−53T2 1 + 13.5iT - 53T^{2}
59 1−10.0T+59T2 1 - 10.0T + 59T^{2}
61 1−7.23T+61T2 1 - 7.23T + 61T^{2}
67 1−2.58iT−67T2 1 - 2.58iT - 67T^{2}
71 1−3.60T+71T2 1 - 3.60T + 71T^{2}
73 1+6.58iT−73T2 1 + 6.58iT - 73T^{2}
79 1−12.4T+79T2 1 - 12.4T + 79T^{2}
83 1−8.68iT−83T2 1 - 8.68iT - 83T^{2}
89 1−6.15T+89T2 1 - 6.15T + 89T^{2}
97 1−2.09iT−97T2 1 - 2.09iT - 97T^{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

−9.414744734026101829189217356741, −8.668727354208290471065811233927, −8.126533496391975444265344838231, −6.82625080765641625841973916194, −6.48233341241629766129742070061, −4.78856400403204038713692570245, −4.23535049240230229610147720438, −3.62025878215266395683639910820, −2.20865440168209913951849714082, −0.992457550403183386729955728712, 0.62756135226311544364931957310, 2.52565330898502292967516854365, 3.77911569648131940269469090822, 4.34082753961749190299647259119, 5.52724213919738405375905769996, 6.34356887320025891768554116049, 7.02990092557533848016824623314, 7.974921935509109493391655900837, 8.531515600231598253713721216823, 9.152358907486455803077493215189

Graph of the ZZ-function along the critical line