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Properties

Label 2-2e3-8.5-c9-0-5
Degree $2$
Conductor $8$
Sign $0.372 + 0.928i$
Analytic cond. $4.12028$
Root an. cond. $2.02984$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.86 + 22.4i)2-s − 247. i·3-s + (−495. + 128. i)4-s − 1.41e3i·5-s + (5.55e3 − 709. i)6-s + 5.08e3·7-s + (−4.31e3 − 1.07e4i)8-s − 4.15e4·9-s + (3.18e4 − 4.06e3i)10-s + 1.48e4i·11-s + (3.18e4 + 1.22e5i)12-s − 6.40e4i·13-s + (1.45e4 + 1.14e5i)14-s − 3.50e5·15-s + (2.28e5 − 1.27e5i)16-s + 2.51e5·17-s + ⋯
L(s)  = 1  + (0.126 + 0.991i)2-s − 1.76i·3-s + (−0.967 + 0.251i)4-s − 1.01i·5-s + (1.74 − 0.223i)6-s + 0.800·7-s + (−0.372 − 0.928i)8-s − 2.10·9-s + (1.00 − 0.128i)10-s + 0.305i·11-s + (0.443 + 1.70i)12-s − 0.622i·13-s + (0.101 + 0.794i)14-s − 1.78·15-s + (0.873 − 0.486i)16-s + 0.729·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.928i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.372 + 0.928i$
Analytic conductor: \(4.12028\)
Root analytic conductor: \(2.02984\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :9/2),\ 0.372 + 0.928i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.14076 - 0.771520i\)
\(L(\frac12)\) \(\approx\) \(1.14076 - 0.771520i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.86 - 22.4i)T \)
good3 \( 1 + 247. iT - 1.96e4T^{2} \)
5 \( 1 + 1.41e3iT - 1.95e6T^{2} \)
7 \( 1 - 5.08e3T + 4.03e7T^{2} \)
11 \( 1 - 1.48e4iT - 2.35e9T^{2} \)
13 \( 1 + 6.40e4iT - 1.06e10T^{2} \)
17 \( 1 - 2.51e5T + 1.18e11T^{2} \)
19 \( 1 - 5.11e5iT - 3.22e11T^{2} \)
23 \( 1 - 1.96e6T + 1.80e12T^{2} \)
29 \( 1 + 2.16e6iT - 1.45e13T^{2} \)
31 \( 1 + 3.03e6T + 2.64e13T^{2} \)
37 \( 1 + 8.74e6iT - 1.29e14T^{2} \)
41 \( 1 + 1.48e7T + 3.27e14T^{2} \)
43 \( 1 + 1.53e5iT - 5.02e14T^{2} \)
47 \( 1 - 5.17e7T + 1.11e15T^{2} \)
53 \( 1 + 4.26e7iT - 3.29e15T^{2} \)
59 \( 1 + 1.15e7iT - 8.66e15T^{2} \)
61 \( 1 - 1.95e8iT - 1.16e16T^{2} \)
67 \( 1 - 1.69e8iT - 2.72e16T^{2} \)
71 \( 1 - 2.23e8T + 4.58e16T^{2} \)
73 \( 1 + 4.10e8T + 5.88e16T^{2} \)
79 \( 1 - 2.51e7T + 1.19e17T^{2} \)
83 \( 1 - 4.73e8iT - 1.86e17T^{2} \)
89 \( 1 - 8.45e8T + 3.50e17T^{2} \)
97 \( 1 + 7.97e8T + 7.60e17T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.96007466625238273116851262794, −17.78133996042136960815267228662, −16.81593882016075729106360703200, −14.62244718858995228697031701352, −13.20262344988885223296115608556, −12.26000126045001234446110840509, −8.539255450859165808688795095002, −7.42601265088147744082964880155, −5.47507876430754981413015897677, −1.02553271768728285703819471725, 3.19765859938353322157558525430, 4.91200434140504340858827066423, 9.066119657328896677898637071310, 10.57606590784860215865387190350, 11.34888803696935809559397074830, 14.19248293819079597878766064704, 15.10227852350045617971139098218, 17.07247912318096439514229388370, 18.75742403191153948371137546485, 20.40902333459441252962047641186

Graph of the $Z$-function along the critical line