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Properties

Label 2-10-5.3-c8-0-3
Degree 22
Conductor 1010
Sign −0.598+0.801i-0.598 + 0.801i
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 + (−39.7 − 39.7i)3-s − 128i·4-s + (−401. − 479. i)5-s − 636.·6-s + (144. − 144. i)7-s + (−1.02e3 − 1.02e3i)8-s − 3.39e3i·9-s + (−7.04e3 − 621. i)10-s + 1.36e4·11-s + (−5.09e3 + 5.09e3i)12-s + (3.08e4 + 3.08e4i)13-s − 2.31e3i·14-s + (−3.09e3 + 3.50e4i)15-s − 1.63e4·16-s + (4.94e3 − 4.94e3i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.491 − 0.491i)3-s − 0.5i·4-s + (−0.642 − 0.766i)5-s − 0.491·6-s + (0.0601 − 0.0601i)7-s + (−0.250 − 0.250i)8-s − 0.517i·9-s + (−0.704 − 0.0621i)10-s + 0.928·11-s + (−0.245 + 0.245i)12-s + (1.08 + 1.08i)13-s − 0.0601i·14-s + (−0.0610 + 0.691i)15-s − 0.250·16-s + (0.0592 − 0.0592i)17-s + ⋯

Functional equation

Λ(s)=(10s/2 ΓC(s) L(s)=((−0.598+0.801i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(10s/2 ΓC(s+4) L(s)=((−0.598+0.801i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1010    =    2⋅52 \cdot 5
Sign: −0.598+0.801i-0.598 + 0.801i
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), −0.598+0.801i)(2,\ 10,\ (\ :4),\ -0.598 + 0.801i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.637158−1.27134i0.637158 - 1.27134i
L(12)L(\frac12) ≈\approx 0.637158−1.27134i0.637158 - 1.27134i
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+(401.+479.i)T 1 + (401. + 479. i)T
good3 1+(39.7+39.7i)T+6.56e3iT2 1 + (39.7 + 39.7i)T + 6.56e3iT^{2}
7 1+(−144.+144.i)T−5.76e6iT2 1 + (-144. + 144. i)T - 5.76e6iT^{2}
11 1−1.36e4T+2.14e8T2 1 - 1.36e4T + 2.14e8T^{2}
13 1+(−3.08e4−3.08e4i)T+8.15e8iT2 1 + (-3.08e4 - 3.08e4i)T + 8.15e8iT^{2}
17 1+(−4.94e3+4.94e3i)T−6.97e9iT2 1 + (-4.94e3 + 4.94e3i)T - 6.97e9iT^{2}
19 1+1.76e5iT−1.69e10T2 1 + 1.76e5iT - 1.69e10T^{2}
23 1+(2.29e5+2.29e5i)T+7.83e10iT2 1 + (2.29e5 + 2.29e5i)T + 7.83e10iT^{2}
29 1+1.44e5iT−5.00e11T2 1 + 1.44e5iT - 5.00e11T^{2}
31 1−1.50e6T+8.52e11T2 1 - 1.50e6T + 8.52e11T^{2}
37 1+(1.60e6−1.60e6i)T−3.51e12iT2 1 + (1.60e6 - 1.60e6i)T - 3.51e12iT^{2}
41 1−3.62e6T+7.98e12T2 1 - 3.62e6T + 7.98e12T^{2}
43 1+(5.75e5+5.75e5i)T+1.16e13iT2 1 + (5.75e5 + 5.75e5i)T + 1.16e13iT^{2}
47 1+(−3.11e6+3.11e6i)T−2.38e13iT2 1 + (-3.11e6 + 3.11e6i)T - 2.38e13iT^{2}
53 1+(−8.16e6−8.16e6i)T+6.22e13iT2 1 + (-8.16e6 - 8.16e6i)T + 6.22e13iT^{2}
59 1+1.54e7iT−1.46e14T2 1 + 1.54e7iT - 1.46e14T^{2}
61 1+1.92e7T+1.91e14T2 1 + 1.92e7T + 1.91e14T^{2}
67 1+(1.00e7−1.00e7i)T−4.06e14iT2 1 + (1.00e7 - 1.00e7i)T - 4.06e14iT^{2}
71 1−1.84e7T+6.45e14T2 1 - 1.84e7T + 6.45e14T^{2}
73 1+(−2.45e7−2.45e7i)T+8.06e14iT2 1 + (-2.45e7 - 2.45e7i)T + 8.06e14iT^{2}
79 1−2.17e7iT−1.51e15T2 1 - 2.17e7iT - 1.51e15T^{2}
83 1+(−4.00e6−4.00e6i)T+2.25e15iT2 1 + (-4.00e6 - 4.00e6i)T + 2.25e15iT^{2}
89 1+6.14e7iT−3.93e15T2 1 + 6.14e7iT - 3.93e15T^{2}
97 1+(−1.55e7+1.55e7i)T−7.83e15iT2 1 + (-1.55e7 + 1.55e7i)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

−18.74899178375951957395398157140, −17.15867894954235647710490256542, −15.62055112811890342928267547497, −13.78214722041872119100607104873, −12.25550210275362855451451000886, −11.42045660839704419276426396280, −8.986740064601155599554704885513, −6.48503423403572714390838654199, −4.21444491938491545766826703369, −0.986585465502458767950738782641, 3.82020611421012658503321505391, 5.96129765775866742918006259161, 7.946550086625866825545912688134, 10.53718074511801605687081798906, 11.92433843335304081474619900785, 13.92637399069533097122386065109, 15.34134078252824709278946876534, 16.35765836656535190721152744493, 17.88928497174011463984732795914, 19.51173591039567534973928063405

Graph of the ZZ-function along the critical line