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Properties

Label 2-10-5.4-c7-0-2
Degree 22
Conductor 1010
Sign 0.526+0.850i0.526 + 0.850i
Analytic cond. 3.123853.12385
Root an. cond. 1.767441.76744
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8i·2-s − 83.6i·3-s − 64·4-s + (237. − 147. i)5-s + 669.·6-s − 185. i·7-s − 512i·8-s − 4.81e3·9-s + (1.17e3 + 1.90e3i)10-s + 3.56e3·11-s + 5.35e3i·12-s + 6.09e3i·13-s + 1.48e3·14-s + (−1.23e4 − 1.98e4i)15-s + 4.09e3·16-s + 1.24e4i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.78i·3-s − 0.5·4-s + (0.850 − 0.526i)5-s + 1.26·6-s − 0.204i·7-s − 0.353i·8-s − 2.20·9-s + (0.371 + 0.601i)10-s + 0.806·11-s + 0.894i·12-s + 0.769i·13-s + 0.144·14-s + (−0.941 − 1.52i)15-s + 0.250·16-s + 0.615i·17-s + ⋯

Functional equation

Λ(s)=(10s/2 ΓC(s) L(s)=((0.526+0.850i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(10s/2 ΓC(s+7/2) L(s)=((0.526+0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1010    =    2⋅52 \cdot 5
Sign: 0.526+0.850i0.526 + 0.850i
Analytic conductor: 3.123853.12385
Root analytic conductor: 1.767441.76744
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ10(9,⋅)\chi_{10} (9, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 10, ( :7/2), 0.526+0.850i)(2,\ 10,\ (\ :7/2),\ 0.526 + 0.850i)

Particular Values

L(4)L(4) ≈\approx 1.25469−0.699239i1.25469 - 0.699239i
L(12)L(\frac12) ≈\approx 1.25469−0.699239i1.25469 - 0.699239i
L(92)L(\frac{9}{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 1−8iT 1 - 8iT
5 1+(−237.+147.i)T 1 + (-237. + 147. i)T
good3 1+83.6iT−2.18e3T2 1 + 83.6iT - 2.18e3T^{2}
7 1+185.iT−8.23e5T2 1 + 185. iT - 8.23e5T^{2}
11 1−3.56e3T+1.94e7T2 1 - 3.56e3T + 1.94e7T^{2}
13 1−6.09e3iT−6.27e7T2 1 - 6.09e3iT - 6.27e7T^{2}
17 1−1.24e4iT−4.10e8T2 1 - 1.24e4iT - 4.10e8T^{2}
19 1−5.06e4T+8.93e8T2 1 - 5.06e4T + 8.93e8T^{2}
23 1−1.14e4iT−3.40e9T2 1 - 1.14e4iT - 3.40e9T^{2}
29 1+1.01e5T+1.72e10T2 1 + 1.01e5T + 1.72e10T^{2}
31 1+2.90e4T+2.75e10T2 1 + 2.90e4T + 2.75e10T^{2}
37 1+1.49e5iT−9.49e10T2 1 + 1.49e5iT - 9.49e10T^{2}
41 1+3.74e5T+1.94e11T2 1 + 3.74e5T + 1.94e11T^{2}
43 1+1.74e5iT−2.71e11T2 1 + 1.74e5iT - 2.71e11T^{2}
47 1−4.28e5iT−5.06e11T2 1 - 4.28e5iT - 5.06e11T^{2}
53 1−1.71e6iT−1.17e12T2 1 - 1.71e6iT - 1.17e12T^{2}
59 1+1.34e5T+2.48e12T2 1 + 1.34e5T + 2.48e12T^{2}
61 1+1.39e6T+3.14e12T2 1 + 1.39e6T + 3.14e12T^{2}
67 1−2.60e6iT−6.06e12T2 1 - 2.60e6iT - 6.06e12T^{2}
71 1+4.91e6T+9.09e12T2 1 + 4.91e6T + 9.09e12T^{2}
73 1+1.19e5iT−1.10e13T2 1 + 1.19e5iT - 1.10e13T^{2}
79 1−4.70e6T+1.92e13T2 1 - 4.70e6T + 1.92e13T^{2}
83 1+9.19e6iT−2.71e13T2 1 + 9.19e6iT - 2.71e13T^{2}
89 1+6.43e6T+4.42e13T2 1 + 6.43e6T + 4.42e13T^{2}
97 1−1.26e7iT−8.07e13T2 1 - 1.26e7iT - 8.07e13T^{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

−18.76133535732387945636910078652, −17.65843574979804520094317654841, −16.73536492962310677015722275597, −14.21426954094864078302826728911, −13.40234642670442655734536262262, −12.04272751746887657229160282832, −9.039626562876661235970843121495, −7.30979794387154382333110495472, −5.93088416668753855450962986916, −1.37003475765328559695814707233, 3.25971690423980935490263412487, 5.32554859408978323330465615091, 9.249043236612006197813301497945, 10.14295639502063941940733858177, 11.47773087953961666413120642771, 13.92559472015997342315228999571, 15.13451449272596670963511007349, 16.71289280521128786009242443137, 18.09722229201436429479026567578, 20.08935962370004503624185608059

Graph of the ZZ-function along the critical line