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Properties

Label 2-10-5.3-c8-0-0
Degree 22
Conductor 1010
Sign −0.783−0.621i-0.783 - 0.621i
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 + (52.9 + 52.9i)3-s − 128i·4-s + (−490. + 387. i)5-s − 847.·6-s + (−2.62e3 + 2.62e3i)7-s + (1.02e3 + 1.02e3i)8-s − 953. i·9-s + (822. − 7.02e3i)10-s + 5.37e3·11-s + (6.77e3 − 6.77e3i)12-s + (2.79e4 + 2.79e4i)13-s − 4.19e4i·14-s + (−4.64e4 − 5.44e3i)15-s − 1.63e4·16-s + (5.34e4 − 5.34e4i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.653 + 0.653i)3-s − 0.5i·4-s + (−0.784 + 0.620i)5-s − 0.653·6-s + (−1.09 + 1.09i)7-s + (0.250 + 0.250i)8-s − 0.145i·9-s + (0.0822 − 0.702i)10-s + 0.366·11-s + (0.326 − 0.326i)12-s + (0.979 + 0.979i)13-s − 1.09i·14-s + (−0.918 − 0.107i)15-s − 0.250·16-s + (0.639 − 0.639i)17-s + ⋯

Functional equation

Λ(s)=(10s/2 ΓC(s) L(s)=((−0.783−0.621i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(10s/2 ΓC(s+4) L(s)=((−0.783−0.621i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1010    =    2⋅52 \cdot 5
Sign: −0.783−0.621i-0.783 - 0.621i
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), −0.783−0.621i)(2,\ 10,\ (\ :4),\ -0.783 - 0.621i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.334505+0.960670i0.334505 + 0.960670i
L(12)L(\frac12) ≈\approx 0.334505+0.960670i0.334505 + 0.960670i
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+(490.−387.i)T 1 + (490. - 387. i)T
good3 1+(−52.9−52.9i)T+6.56e3iT2 1 + (-52.9 - 52.9i)T + 6.56e3iT^{2}
7 1+(2.62e3−2.62e3i)T−5.76e6iT2 1 + (2.62e3 - 2.62e3i)T - 5.76e6iT^{2}
11 1−5.37e3T+2.14e8T2 1 - 5.37e3T + 2.14e8T^{2}
13 1+(−2.79e4−2.79e4i)T+8.15e8iT2 1 + (-2.79e4 - 2.79e4i)T + 8.15e8iT^{2}
17 1+(−5.34e4+5.34e4i)T−6.97e9iT2 1 + (-5.34e4 + 5.34e4i)T - 6.97e9iT^{2}
19 1−1.44e5iT−1.69e10T2 1 - 1.44e5iT - 1.69e10T^{2}
23 1+(−5.21e4−5.21e4i)T+7.83e10iT2 1 + (-5.21e4 - 5.21e4i)T + 7.83e10iT^{2}
29 1−4.17e4iT−5.00e11T2 1 - 4.17e4iT - 5.00e11T^{2}
31 1−2.44e5T+8.52e11T2 1 - 2.44e5T + 8.52e11T^{2}
37 1+(2.00e6−2.00e6i)T−3.51e12iT2 1 + (2.00e6 - 2.00e6i)T - 3.51e12iT^{2}
41 1+4.17e6T+7.98e12T2 1 + 4.17e6T + 7.98e12T^{2}
43 1+(−4.01e6−4.01e6i)T+1.16e13iT2 1 + (-4.01e6 - 4.01e6i)T + 1.16e13iT^{2}
47 1+(−2.26e6+2.26e6i)T−2.38e13iT2 1 + (-2.26e6 + 2.26e6i)T - 2.38e13iT^{2}
53 1+(3.72e6+3.72e6i)T+6.22e13iT2 1 + (3.72e6 + 3.72e6i)T + 6.22e13iT^{2}
59 1+1.12e7iT−1.46e14T2 1 + 1.12e7iT - 1.46e14T^{2}
61 1−2.02e7T+1.91e14T2 1 - 2.02e7T + 1.91e14T^{2}
67 1+(1.30e7−1.30e7i)T−4.06e14iT2 1 + (1.30e7 - 1.30e7i)T - 4.06e14iT^{2}
71 1−2.61e7T+6.45e14T2 1 - 2.61e7T + 6.45e14T^{2}
73 1+(1.85e7+1.85e7i)T+8.06e14iT2 1 + (1.85e7 + 1.85e7i)T + 8.06e14iT^{2}
79 1−3.07e6iT−1.51e15T2 1 - 3.07e6iT - 1.51e15T^{2}
83 1+(−2.41e7−2.41e7i)T+2.25e15iT2 1 + (-2.41e7 - 2.41e7i)T + 2.25e15iT^{2}
89 1+5.76e6iT−3.93e15T2 1 + 5.76e6iT - 3.93e15T^{2}
97 1+(−2.86e7+2.86e7i)T−7.83e15iT2 1 + (-2.86e7 + 2.86e7i)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

−19.22039782453345498122489228997, −18.55225175777111055126870650811, −16.25195716448355735395762076240, −15.49118348210720808768659522907, −14.30501672796265016763968391433, −11.88678472185384620331631544692, −9.815915125807387521427744484699, −8.610619082343812331927088025774, −6.47798515963746562097895777788, −3.43396587058692779369520740118, 0.801098543784933523304306953121, 3.52661646363352644481874717553, 7.31654700935727973072703631691, 8.687319578953972657443850994476, 10.62947254230871342329311705112, 12.62228437719150411911918321604, 13.54302968260372092947213546653, 15.85370592360411122446267774804, 17.13582691403347046369645163075, 19.01291095096740593356407572430

Graph of the ZZ-function along the critical line