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Properties

Label 2-10-5.3-c8-0-2
Degree 22
Conductor 1010
Sign 0.699+0.715i0.699 + 0.715i
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 + (−25.9 − 25.9i)3-s − 128i·4-s + (535. − 322. i)5-s + 415.·6-s + (2.03e3 − 2.03e3i)7-s + (1.02e3 + 1.02e3i)8-s − 5.21e3i·9-s + (−1.70e3 + 6.86e3i)10-s − 1.52e4·11-s + (−3.32e3 + 3.32e3i)12-s + (8.87e3 + 8.87e3i)13-s + 3.24e4i·14-s + (−2.22e4 − 5.52e3i)15-s − 1.63e4·16-s + (4.60e4 − 4.60e4i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.320 − 0.320i)3-s − 0.5i·4-s + (0.856 − 0.516i)5-s + 0.320·6-s + (0.845 − 0.845i)7-s + (0.250 + 0.250i)8-s − 0.794i·9-s + (−0.170 + 0.686i)10-s − 1.04·11-s + (−0.160 + 0.160i)12-s + (0.310 + 0.310i)13-s + 0.845i·14-s + (−0.439 − 0.109i)15-s − 0.250·16-s + (0.551 − 0.551i)17-s + ⋯

Functional equation

Λ(s)=(10s/2 ΓC(s) L(s)=((0.699+0.715i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.715i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(10s/2 ΓC(s+4) L(s)=((0.699+0.715i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.699 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1010    =    2⋅52 \cdot 5
Sign: 0.699+0.715i0.699 + 0.715i
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), 0.699+0.715i)(2,\ 10,\ (\ :4),\ 0.699 + 0.715i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.08087−0.454895i1.08087 - 0.454895i
L(12)L(\frac12) ≈\approx 1.08087−0.454895i1.08087 - 0.454895i
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+(−535.+322.i)T 1 + (-535. + 322. i)T
good3 1+(25.9+25.9i)T+6.56e3iT2 1 + (25.9 + 25.9i)T + 6.56e3iT^{2}
7 1+(−2.03e3+2.03e3i)T−5.76e6iT2 1 + (-2.03e3 + 2.03e3i)T - 5.76e6iT^{2}
11 1+1.52e4T+2.14e8T2 1 + 1.52e4T + 2.14e8T^{2}
13 1+(−8.87e3−8.87e3i)T+8.15e8iT2 1 + (-8.87e3 - 8.87e3i)T + 8.15e8iT^{2}
17 1+(−4.60e4+4.60e4i)T−6.97e9iT2 1 + (-4.60e4 + 4.60e4i)T - 6.97e9iT^{2}
19 1+7.32e4iT−1.69e10T2 1 + 7.32e4iT - 1.69e10T^{2}
23 1+(−2.63e5−2.63e5i)T+7.83e10iT2 1 + (-2.63e5 - 2.63e5i)T + 7.83e10iT^{2}
29 1−1.20e6iT−5.00e11T2 1 - 1.20e6iT - 5.00e11T^{2}
31 1+1.40e6T+8.52e11T2 1 + 1.40e6T + 8.52e11T^{2}
37 1+(−1.51e6+1.51e6i)T−3.51e12iT2 1 + (-1.51e6 + 1.51e6i)T - 3.51e12iT^{2}
41 1−2.03e6T+7.98e12T2 1 - 2.03e6T + 7.98e12T^{2}
43 1+(4.41e5+4.41e5i)T+1.16e13iT2 1 + (4.41e5 + 4.41e5i)T + 1.16e13iT^{2}
47 1+(3.91e6−3.91e6i)T−2.38e13iT2 1 + (3.91e6 - 3.91e6i)T - 2.38e13iT^{2}
53 1+(−8.72e6−8.72e6i)T+6.22e13iT2 1 + (-8.72e6 - 8.72e6i)T + 6.22e13iT^{2}
59 1+4.77e6iT−1.46e14T2 1 + 4.77e6iT - 1.46e14T^{2}
61 1−2.24e7T+1.91e14T2 1 - 2.24e7T + 1.91e14T^{2}
67 1+(1.55e7−1.55e7i)T−4.06e14iT2 1 + (1.55e7 - 1.55e7i)T - 4.06e14iT^{2}
71 1−3.03e7T+6.45e14T2 1 - 3.03e7T + 6.45e14T^{2}
73 1+(1.18e7+1.18e7i)T+8.06e14iT2 1 + (1.18e7 + 1.18e7i)T + 8.06e14iT^{2}
79 1+2.53e7iT−1.51e15T2 1 + 2.53e7iT - 1.51e15T^{2}
83 1+(−1.13e7−1.13e7i)T+2.25e15iT2 1 + (-1.13e7 - 1.13e7i)T + 2.25e15iT^{2}
89 1−1.02e7iT−3.93e15T2 1 - 1.02e7iT - 3.93e15T^{2}
97 1+(−4.11e7+4.11e7i)T−7.83e15iT2 1 + (-4.11e7 + 4.11e7i)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

−18.26956493250383704184179346780, −17.57676066475171979043589777511, −16.34558823557599653074846658234, −14.48880283528666747278736147216, −13.05163793567470218738733518796, −10.94366761240132015743559132895, −9.209487462885946197681305327800, −7.29154380092448956159123600654, −5.34262322075842697755823825540, −1.09122246047220759063044200525, 2.25191905571300630546010570399, 5.45269018885786286140952283594, 8.154952320465890285594692637568, 10.12109764209585831458647257869, 11.19450287195086172685431149576, 13.11309136458685725931638834512, 14.91159500540444449651290469259, 16.66697653822403080057316495135, 18.02987507352447478999831013558, 18.85516239329005465625546631500

Graph of the ZZ-function along the critical line