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Properties

Label 2-10-5.3-c8-0-1
Degree 22
Conductor 1010
Sign 0.984−0.177i0.984 - 0.177i
Analytic cond. 4.073784.07378
Root an. cond. 2.018362.01836
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8 − 8i)2-s + (82.7 + 82.7i)3-s − 128i·4-s + (−33.6 + 624. i)5-s + 1.32e3·6-s + (2.71e3 − 2.71e3i)7-s + (−1.02e3 − 1.02e3i)8-s + 7.14e3i·9-s + (4.72e3 + 5.26e3i)10-s − 2.09e4·11-s + (1.05e4 − 1.05e4i)12-s + (−8.09e3 − 8.09e3i)13-s − 4.34e4i·14-s + (−5.44e4 + 4.88e4i)15-s − 1.63e4·16-s + (−3.14e3 + 3.14e3i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (1.02 + 1.02i)3-s − 0.5i·4-s + (−0.0538 + 0.998i)5-s + 1.02·6-s + (1.13 − 1.13i)7-s + (−0.250 − 0.250i)8-s + 1.08i·9-s + (0.472 + 0.526i)10-s − 1.43·11-s + (0.511 − 0.511i)12-s + (−0.283 − 0.283i)13-s − 1.13i·14-s + (−1.07 + 0.965i)15-s − 0.250·16-s + (−0.0376 + 0.0376i)17-s + ⋯

Functional equation

Λ(s)=(10s/2 ΓC(s) L(s)=((0.984−0.177i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(10s/2 ΓC(s+4) L(s)=((0.984−0.177i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1010    =    2⋅52 \cdot 5
Sign: 0.984−0.177i0.984 - 0.177i
Analytic conductor: 4.073784.07378
Root analytic conductor: 2.018362.01836
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ10(3,⋅)\chi_{10} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 10, ( :4), 0.984−0.177i)(2,\ 10,\ (\ :4),\ 0.984 - 0.177i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 2.43627+0.217379i2.43627 + 0.217379i
L(12)L(\frac12) ≈\approx 2.43627+0.217379i2.43627 + 0.217379i
L(5)L(5) 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+(−8+8i)T 1 + (-8 + 8i)T
5 1+(33.6−624.i)T 1 + (33.6 - 624. i)T
good3 1+(−82.7−82.7i)T+6.56e3iT2 1 + (-82.7 - 82.7i)T + 6.56e3iT^{2}
7 1+(−2.71e3+2.71e3i)T−5.76e6iT2 1 + (-2.71e3 + 2.71e3i)T - 5.76e6iT^{2}
11 1+2.09e4T+2.14e8T2 1 + 2.09e4T + 2.14e8T^{2}
13 1+(8.09e3+8.09e3i)T+8.15e8iT2 1 + (8.09e3 + 8.09e3i)T + 8.15e8iT^{2}
17 1+(3.14e3−3.14e3i)T−6.97e9iT2 1 + (3.14e3 - 3.14e3i)T - 6.97e9iT^{2}
19 1+6.92e4iT−1.69e10T2 1 + 6.92e4iT - 1.69e10T^{2}
23 1+(−1.08e3−1.08e3i)T+7.83e10iT2 1 + (-1.08e3 - 1.08e3i)T + 7.83e10iT^{2}
29 1+1.20e5iT−5.00e11T2 1 + 1.20e5iT - 5.00e11T^{2}
31 1−8.26e5T+8.52e11T2 1 - 8.26e5T + 8.52e11T^{2}
37 1+(1.17e6−1.17e6i)T−3.51e12iT2 1 + (1.17e6 - 1.17e6i)T - 3.51e12iT^{2}
41 1−1.74e6T+7.98e12T2 1 - 1.74e6T + 7.98e12T^{2}
43 1+(−3.03e6−3.03e6i)T+1.16e13iT2 1 + (-3.03e6 - 3.03e6i)T + 1.16e13iT^{2}
47 1+(5.83e6−5.83e6i)T−2.38e13iT2 1 + (5.83e6 - 5.83e6i)T - 2.38e13iT^{2}
53 1+(−1.97e6−1.97e6i)T+6.22e13iT2 1 + (-1.97e6 - 1.97e6i)T + 6.22e13iT^{2}
59 1−1.58e7iT−1.46e14T2 1 - 1.58e7iT - 1.46e14T^{2}
61 1+2.51e6T+1.91e14T2 1 + 2.51e6T + 1.91e14T^{2}
67 1+(−1.47e7+1.47e7i)T−4.06e14iT2 1 + (-1.47e7 + 1.47e7i)T - 4.06e14iT^{2}
71 1+8.22e6T+6.45e14T2 1 + 8.22e6T + 6.45e14T^{2}
73 1+(3.03e7+3.03e7i)T+8.06e14iT2 1 + (3.03e7 + 3.03e7i)T + 8.06e14iT^{2}
79 1−4.69e7iT−1.51e15T2 1 - 4.69e7iT - 1.51e15T^{2}
83 1+(−2.81e7−2.81e7i)T+2.25e15iT2 1 + (-2.81e7 - 2.81e7i)T + 2.25e15iT^{2}
89 1+7.91e7iT−3.93e15T2 1 + 7.91e7iT - 3.93e15T^{2}
97 1+(−1.47e6+1.47e6i)T−7.83e15iT2 1 + (-1.47e6 + 1.47e6i)T - 7.83e15iT^{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

−19.54042629232839871684162640187, −17.85750253085818060856408635700, −15.58624105706325791902748358787, −14.58615806482881108838494555425, −13.62620102160974074703749549077, −10.97466073515949477971240202201, −10.12407096360998248948078790266, −7.81725539377555679860902636651, −4.53669008119461396286520916113, −2.84166717444066003911303309946, 2.16456074157436581167961862165, 5.22986146145030304615417834263, 7.81246732371750747120041890220, 8.648420345736611399246963327099, 12.09895903757120815560541844909, 13.16067869683265866034452964177, 14.46442209357481424235818528492, 15.79063758485814265398196514834, 17.74287176166650237656429495845, 18.90421261364371334819173431548

Graph of the ZZ-function along the critical line