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Properties

Label 2-720-20.19-c2-0-15
Degree 22
Conductor 720720
Sign 0.948−0.316i0.948 - 0.316i
Analytic cond. 19.618519.6185
Root an. cond. 4.429284.42928
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 4.89i)5-s + 8.48·7-s + 13.8i·11-s + 9.79i·13-s + 19.5i·17-s + 13.8i·19-s + 25.4·23-s + (−22.9 − 9.79i)25-s + 22·29-s − 55.4i·31-s + (8.48 − 41.5i)35-s + 48.9i·37-s − 22·41-s + 59.3·43-s − 8.48·47-s + ⋯
L(s)  = 1  + (0.200 − 0.979i)5-s + 1.21·7-s + 1.25i·11-s + 0.753i·13-s + 1.15i·17-s + 0.729i·19-s + 1.10·23-s + (−0.919 − 0.391i)25-s + 0.758·29-s − 1.78i·31-s + (0.242 − 1.18i)35-s + 1.32i·37-s − 0.536·41-s + 1.38·43-s − 0.180·47-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.948−0.316i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1) L(s)=((0.948−0.316i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.948−0.316i0.948 - 0.316i
Analytic conductor: 19.618519.6185
Root analytic conductor: 4.429284.42928
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ720(559,⋅)\chi_{720} (559, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1), 0.948−0.316i)(2,\ 720,\ (\ :1),\ 0.948 - 0.316i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.1991000842.199100084
L(12)L(\frac12) ≈\approx 2.1991000842.199100084
L(2)L(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 1
5 1+(−1+4.89i)T 1 + (-1 + 4.89i)T
good7 1−8.48T+49T2 1 - 8.48T + 49T^{2}
11 1−13.8iT−121T2 1 - 13.8iT - 121T^{2}
13 1−9.79iT−169T2 1 - 9.79iT - 169T^{2}
17 1−19.5iT−289T2 1 - 19.5iT - 289T^{2}
19 1−13.8iT−361T2 1 - 13.8iT - 361T^{2}
23 1−25.4T+529T2 1 - 25.4T + 529T^{2}
29 1−22T+841T2 1 - 22T + 841T^{2}
31 1+55.4iT−961T2 1 + 55.4iT - 961T^{2}
37 1−48.9iT−1.36e3T2 1 - 48.9iT - 1.36e3T^{2}
41 1+22T+1.68e3T2 1 + 22T + 1.68e3T^{2}
43 1−59.3T+1.84e3T2 1 - 59.3T + 1.84e3T^{2}
47 1+8.48T+2.20e3T2 1 + 8.48T + 2.20e3T^{2}
53 1+29.3iT−2.80e3T2 1 + 29.3iT - 2.80e3T^{2}
59 1−13.8iT−3.48e3T2 1 - 13.8iT - 3.48e3T^{2}
61 1−46T+3.72e3T2 1 - 46T + 3.72e3T^{2}
67 1−59.3T+4.48e3T2 1 - 59.3T + 4.48e3T^{2}
71 1+27.7iT−5.04e3T2 1 + 27.7iT - 5.04e3T^{2}
73 1−78.3iT−5.32e3T2 1 - 78.3iT - 5.32e3T^{2}
79 1−6.24e3T2 1 - 6.24e3T^{2}
83 1−76.3T+6.88e3T2 1 - 76.3T + 6.88e3T^{2}
89 1+146T+7.92e3T2 1 + 146T + 7.92e3T^{2}
97 1+58.7iT−9.40e3T2 1 + 58.7iT - 9.40e3T^{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

−10.12314444933941136286159186327, −9.408567079873112220094318648353, −8.417086141303152316892211975458, −7.896748022563885638265485421799, −6.76587457971949025501721229378, −5.61012994713665419596218429139, −4.67943389130162563053487187332, −4.12173243985739374354569372247, −2.12328433888116365851164971099, −1.30989940018502029144559718886, 0.898934350621925357743928838947, 2.54929407515441619522061153894, 3.36009374717947888408977617960, 4.86907727573428887911347332534, 5.59047393734977297073220558731, 6.75674326683683776189357233670, 7.50687693131414763856887119901, 8.439404926580497552975778004389, 9.213650667793910855640697119645, 10.46104089603655875300388917027

Graph of the ZZ-function along the critical line