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Properties

Label 2-720-9.5-c2-0-42
Degree 22
Conductor 720720
Sign −0.818+0.574i-0.818 + 0.574i
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  + (0.129 − 2.99i)3-s + (1.93 + 1.11i)5-s + (−3.31 − 5.74i)7-s + (−8.96 − 0.774i)9-s + (18.2 − 10.5i)11-s + (4.89 − 8.47i)13-s + (3.60 − 5.65i)15-s + 12.9i·17-s − 16.1·19-s + (−17.6 + 9.19i)21-s + (−14.6 − 8.46i)23-s + (2.5 + 4.33i)25-s + (−3.48 + 26.7i)27-s + (43.3 − 25.0i)29-s + (−5.03 + 8.71i)31-s + ⋯
L(s)  = 1  + (0.0430 − 0.999i)3-s + (0.387 + 0.223i)5-s + (−0.473 − 0.820i)7-s + (−0.996 − 0.0860i)9-s + (1.65 − 0.956i)11-s + (0.376 − 0.651i)13-s + (0.240 − 0.377i)15-s + 0.758i·17-s − 0.851·19-s + (−0.840 + 0.437i)21-s + (−0.637 − 0.368i)23-s + (0.100 + 0.173i)25-s + (−0.128 + 0.991i)27-s + (1.49 − 0.863i)29-s + (−0.162 + 0.281i)31-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((−0.818+0.574i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s+1) L(s)=((−0.818+0.574i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: −0.818+0.574i-0.818 + 0.574i
Analytic conductor: 19.618519.6185
Root analytic conductor: 4.429284.42928
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ720(401,⋅)\chi_{720} (401, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :1), −0.818+0.574i)(2,\ 720,\ (\ :1),\ -0.818 + 0.574i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.6057498721.605749872
L(12)L(\frac12) ≈\approx 1.6057498721.605749872
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+(−0.129+2.99i)T 1 + (-0.129 + 2.99i)T
5 1+(−1.93−1.11i)T 1 + (-1.93 - 1.11i)T
good7 1+(3.31+5.74i)T+(−24.5+42.4i)T2 1 + (3.31 + 5.74i)T + (-24.5 + 42.4i)T^{2}
11 1+(−18.2+10.5i)T+(60.5−104.i)T2 1 + (-18.2 + 10.5i)T + (60.5 - 104. i)T^{2}
13 1+(−4.89+8.47i)T+(−84.5−146.i)T2 1 + (-4.89 + 8.47i)T + (-84.5 - 146. i)T^{2}
17 1−12.9iT−289T2 1 - 12.9iT - 289T^{2}
19 1+16.1T+361T2 1 + 16.1T + 361T^{2}
23 1+(14.6+8.46i)T+(264.5+458.i)T2 1 + (14.6 + 8.46i)T + (264.5 + 458. i)T^{2}
29 1+(−43.3+25.0i)T+(420.5−728.i)T2 1 + (-43.3 + 25.0i)T + (420.5 - 728. i)T^{2}
31 1+(5.03−8.71i)T+(−480.5−832.i)T2 1 + (5.03 - 8.71i)T + (-480.5 - 832. i)T^{2}
37 1−2.54T+1.36e3T2 1 - 2.54T + 1.36e3T^{2}
41 1+(46.5+26.8i)T+(840.5+1.45e3i)T2 1 + (46.5 + 26.8i)T + (840.5 + 1.45e3i)T^{2}
43 1+(37.4+64.8i)T+(−924.5+1.60e3i)T2 1 + (37.4 + 64.8i)T + (-924.5 + 1.60e3i)T^{2}
47 1+(40.8−23.6i)T+(1.10e3−1.91e3i)T2 1 + (40.8 - 23.6i)T + (1.10e3 - 1.91e3i)T^{2}
53 1−1.08iT−2.80e3T2 1 - 1.08iT - 2.80e3T^{2}
59 1+(8.34+4.81i)T+(1.74e3+3.01e3i)T2 1 + (8.34 + 4.81i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(−22.4−38.8i)T+(−1.86e3+3.22e3i)T2 1 + (-22.4 - 38.8i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(1.67−2.89i)T+(−2.24e3−3.88e3i)T2 1 + (1.67 - 2.89i)T + (-2.24e3 - 3.88e3i)T^{2}
71 1+38.4iT−5.04e3T2 1 + 38.4iT - 5.04e3T^{2}
73 1+88.9T+5.32e3T2 1 + 88.9T + 5.32e3T^{2}
79 1+(−58.8−101.i)T+(−3.12e3+5.40e3i)T2 1 + (-58.8 - 101. i)T + (-3.12e3 + 5.40e3i)T^{2}
83 1+(127.−73.5i)T+(3.44e3−5.96e3i)T2 1 + (127. - 73.5i)T + (3.44e3 - 5.96e3i)T^{2}
89 1+145.iT−7.92e3T2 1 + 145. iT - 7.92e3T^{2}
97 1+(−23.5−40.7i)T+(−4.70e3+8.14e3i)T2 1 + (-23.5 - 40.7i)T + (-4.70e3 + 8.14e3i)T^{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

−9.994674367092882778171375369261, −8.672088341794956159377359897239, −8.318339883226714532866169245007, −6.94695366614190163135171237369, −6.45318636900229914904056349116, −5.78826725802268909879233145543, −4.05658519325733717037529804119, −3.17905924791495124672416443801, −1.70545466415228159975940191932, −0.56408307043847950573352941434, 1.73900164647814811790676569679, 3.07666235332492458586447772769, 4.22541744651888556896162067844, 4.95317175698434017063410641919, 6.19774084616650980631005259708, 6.71148956959891407655432550652, 8.358822890762445117212585934715, 9.091385903485710446976548123385, 9.578212116319932515725450888512, 10.27230129900154725394824505834

Graph of the ZZ-function along the critical line