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Properties

Label 2-720-15.14-c2-0-9
Degree 22
Conductor 720720
Sign 0.998−0.0519i0.998 - 0.0519i
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  + (−3.09 − 3.92i)5-s + 0.822i·7-s + 5.93i·11-s + 11.9i·13-s − 14.0·17-s + 21.5·19-s + 26.9·23-s + (−5.84 + 24.3i)25-s − 26.9i·29-s + 27.3·31-s + (3.23 − 2.54i)35-s + 10.8i·37-s − 47.5i·41-s − 19.0i·43-s + 89.1·47-s + ⋯
L(s)  = 1  + (−0.618 − 0.785i)5-s + 0.117i·7-s + 0.539i·11-s + 0.917i·13-s − 0.824·17-s + 1.13·19-s + 1.17·23-s + (−0.233 + 0.972i)25-s − 0.929i·29-s + 0.881·31-s + (0.0923 − 0.0727i)35-s + 0.294i·37-s − 1.15i·41-s − 0.443i·43-s + 1.89·47-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.998−0.0519i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0519i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1) L(s)=((0.998−0.0519i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.998−0.0519i0.998 - 0.0519i
Analytic conductor: 19.618519.6185
Root analytic conductor: 4.429284.42928
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ720(449,⋅)\chi_{720} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1), 0.998−0.0519i)(2,\ 720,\ (\ :1),\ 0.998 - 0.0519i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.5697331381.569733138
L(12)L(\frac12) ≈\approx 1.5697331381.569733138
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+(3.09+3.92i)T 1 + (3.09 + 3.92i)T
good7 1−0.822iT−49T2 1 - 0.822iT - 49T^{2}
11 1−5.93iT−121T2 1 - 5.93iT - 121T^{2}
13 1−11.9iT−169T2 1 - 11.9iT - 169T^{2}
17 1+14.0T+289T2 1 + 14.0T + 289T^{2}
19 1−21.5T+361T2 1 - 21.5T + 361T^{2}
23 1−26.9T+529T2 1 - 26.9T + 529T^{2}
29 1+26.9iT−841T2 1 + 26.9iT - 841T^{2}
31 1−27.3T+961T2 1 - 27.3T + 961T^{2}
37 1−10.8iT−1.36e3T2 1 - 10.8iT - 1.36e3T^{2}
41 1+47.5iT−1.68e3T2 1 + 47.5iT - 1.68e3T^{2}
43 1+19.0iT−1.84e3T2 1 + 19.0iT - 1.84e3T^{2}
47 1−89.1T+2.20e3T2 1 - 89.1T + 2.20e3T^{2}
53 1−15.3T+2.80e3T2 1 - 15.3T + 2.80e3T^{2}
59 1−15.6iT−3.48e3T2 1 - 15.6iT - 3.48e3T^{2}
61 1−17.6T+3.72e3T2 1 - 17.6T + 3.72e3T^{2}
67 1−116.iT−4.48e3T2 1 - 116. iT - 4.48e3T^{2}
71 1+84.4iT−5.04e3T2 1 + 84.4iT - 5.04e3T^{2}
73 1−79.0iT−5.32e3T2 1 - 79.0iT - 5.32e3T^{2}
79 1−94.9T+6.24e3T2 1 - 94.9T + 6.24e3T^{2}
83 1+74.3T+6.88e3T2 1 + 74.3T + 6.88e3T^{2}
89 1−90.7iT−7.92e3T2 1 - 90.7iT - 7.92e3T^{2}
97 1+141.iT−9.40e3T2 1 + 141. iT - 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

−10.13299449517919735763896037668, −9.148022575487812149388333250825, −8.704908062989409847858659370200, −7.53106829327361347363541970549, −6.90607273951288477838107447966, −5.59362378847504304698677553401, −4.65437814534231219369086321913, −3.89092625803490187464202616929, −2.39104390845829924636407095541, −0.917947560587468141852848191221, 0.78855771578154660031043502641, 2.72830620126721884604052081189, 3.45716111607789966458649552428, 4.69014477437059284876380755234, 5.78272888424273278275920383573, 6.82084200390352461621662704058, 7.53674214462236273691808541882, 8.396885837640545987207775679280, 9.331289755336329138038926427928, 10.43763479122412624939899521619

Graph of the ZZ-function along the critical line