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Properties

Label 2-864-9.2-c2-0-2
Degree 22
Conductor 864864
Sign −0.780−0.625i-0.780 - 0.625i
Analytic cond. 23.542223.5422
Root an. cond. 4.852044.85204
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  + (6.30 − 3.63i)5-s + (−4.46 + 7.74i)7-s + (−10.5 − 6.08i)11-s + (−1.73 − 3.00i)13-s + 33.3i·17-s − 33.3·19-s + (−15.8 + 9.16i)23-s + (13.9 − 24.2i)25-s + (−13.6 − 7.85i)29-s + (6.66 + 11.5i)31-s + 65.0i·35-s − 35.3·37-s + (−4.55 + 2.62i)41-s + (20.1 − 34.9i)43-s + (29.9 + 17.2i)47-s + ⋯
L(s)  = 1  + (1.26 − 0.727i)5-s + (−0.638 + 1.10i)7-s + (−0.958 − 0.553i)11-s + (−0.133 − 0.231i)13-s + 1.95i·17-s − 1.75·19-s + (−0.690 + 0.398i)23-s + (0.558 − 0.968i)25-s + (−0.469 − 0.270i)29-s + (0.215 + 0.372i)31-s + 1.85i·35-s − 0.956·37-s + (−0.111 + 0.0640i)41-s + (0.469 − 0.812i)43-s + (0.637 + 0.367i)47-s + ⋯

Functional equation

Λ(s)=(864s/2 ΓC(s) L(s)=((−0.780−0.625i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(864s/2 ΓC(s+1) L(s)=((−0.780−0.625i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 864864    =    25⋅332^{5} \cdot 3^{3}
Sign: −0.780−0.625i-0.780 - 0.625i
Analytic conductor: 23.542223.5422
Root analytic conductor: 4.852044.85204
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ864(737,⋅)\chi_{864} (737, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 864, ( :1), −0.780−0.625i)(2,\ 864,\ (\ :1),\ -0.780 - 0.625i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.66068876120.6606887612
L(12)L(\frac12) ≈\approx 0.66068876120.6606887612
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
good5 1+(−6.30+3.63i)T+(12.5−21.6i)T2 1 + (-6.30 + 3.63i)T + (12.5 - 21.6i)T^{2}
7 1+(4.46−7.74i)T+(−24.5−42.4i)T2 1 + (4.46 - 7.74i)T + (-24.5 - 42.4i)T^{2}
11 1+(10.5+6.08i)T+(60.5+104.i)T2 1 + (10.5 + 6.08i)T + (60.5 + 104. i)T^{2}
13 1+(1.73+3.00i)T+(−84.5+146.i)T2 1 + (1.73 + 3.00i)T + (-84.5 + 146. i)T^{2}
17 1−33.3iT−289T2 1 - 33.3iT - 289T^{2}
19 1+33.3T+361T2 1 + 33.3T + 361T^{2}
23 1+(15.8−9.16i)T+(264.5−458.i)T2 1 + (15.8 - 9.16i)T + (264.5 - 458. i)T^{2}
29 1+(13.6+7.85i)T+(420.5+728.i)T2 1 + (13.6 + 7.85i)T + (420.5 + 728. i)T^{2}
31 1+(−6.66−11.5i)T+(−480.5+832.i)T2 1 + (-6.66 - 11.5i)T + (-480.5 + 832. i)T^{2}
37 1+35.3T+1.36e3T2 1 + 35.3T + 1.36e3T^{2}
41 1+(4.55−2.62i)T+(840.5−1.45e3i)T2 1 + (4.55 - 2.62i)T + (840.5 - 1.45e3i)T^{2}
43 1+(−20.1+34.9i)T+(−924.5−1.60e3i)T2 1 + (-20.1 + 34.9i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(−29.9−17.2i)T+(1.10e3+1.91e3i)T2 1 + (-29.9 - 17.2i)T + (1.10e3 + 1.91e3i)T^{2}
53 1−15.9iT−2.80e3T2 1 - 15.9iT - 2.80e3T^{2}
59 1+(16.8−9.70i)T+(1.74e3−3.01e3i)T2 1 + (16.8 - 9.70i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(−7.12+12.3i)T+(−1.86e3−3.22e3i)T2 1 + (-7.12 + 12.3i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(16.8+29.1i)T+(−2.24e3+3.88e3i)T2 1 + (16.8 + 29.1i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1−21.5iT−5.04e3T2 1 - 21.5iT - 5.04e3T^{2}
73 1−35.0T+5.32e3T2 1 - 35.0T + 5.32e3T^{2}
79 1+(75.5−130.i)T+(−3.12e3−5.40e3i)T2 1 + (75.5 - 130. i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−107.−62.2i)T+(3.44e3+5.96e3i)T2 1 + (-107. - 62.2i)T + (3.44e3 + 5.96e3i)T^{2}
89 1+58.9iT−7.92e3T2 1 + 58.9iT - 7.92e3T^{2}
97 1+(43.2−74.8i)T+(−4.70e3−8.14e3i)T2 1 + (43.2 - 74.8i)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

−10.31117237018221523209813143524, −9.384832201276182837204016794665, −8.643939636870364896450484333666, −8.125988346037981950058721816403, −6.47755416343697063047679761894, −5.83243667668603924916652936961, −5.37783260039226569598165938187, −3.97731428877827932983276341526, −2.55843662476502723451462132694, −1.76862002841907408786818491462, 0.19123558052287117443161918066, 2.07218503957746347984291045877, 2.86204218801214152385674997793, 4.22283146370193264669031610602, 5.24555622566095279425613842563, 6.35490399474953389957909821683, 6.93535374359344082217309029254, 7.69717631048480851743682868550, 9.057280178929481940571223749055, 9.855177543568564516837001011204

Graph of the ZZ-function along the critical line