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Properties

Label 2-2e2-4.3-c18-0-0
Degree $2$
Conductor $4$
Sign $0.671 - 0.741i$
Analytic cond. $8.21544$
Root an. cond. $2.86625$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−207. − 468. i)2-s − 4.12e3i·3-s + (−1.75e5 + 1.94e5i)4-s − 1.32e6·5-s + (−1.93e6 + 8.56e5i)6-s + 1.88e7i·7-s + (1.27e8 + 4.20e7i)8-s + 3.70e8·9-s + (2.74e8 + 6.18e8i)10-s + 2.94e9i·11-s + (8.02e8 + 7.26e8i)12-s + 2.06e9·13-s + (8.84e9 − 3.92e9i)14-s + 5.45e9i·15-s + (−6.79e9 − 6.83e10i)16-s − 1.59e11·17-s + ⋯
L(s)  = 1  + (−0.405 − 0.914i)2-s − 0.209i·3-s + (−0.671 + 0.741i)4-s − 0.676·5-s + (−0.191 + 0.0850i)6-s + 0.468i·7-s + (0.949 + 0.313i)8-s + 0.956·9-s + (0.274 + 0.618i)10-s + 1.24i·11-s + (0.155 + 0.140i)12-s + 0.194·13-s + (0.427 − 0.189i)14-s + 0.141i·15-s + (−0.0988 − 0.995i)16-s − 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.671 - 0.741i$
Analytic conductor: \(8.21544\)
Root analytic conductor: \(2.86625\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :9),\ 0.671 - 0.741i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.736972 + 0.326864i\)
\(L(\frac12)\) \(\approx\) \(0.736972 + 0.326864i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (207. + 468. i)T \)
good3 \( 1 + 4.12e3iT - 3.87e8T^{2} \)
5 \( 1 + 1.32e6T + 3.81e12T^{2} \)
7 \( 1 - 1.88e7iT - 1.62e15T^{2} \)
11 \( 1 - 2.94e9iT - 5.55e18T^{2} \)
13 \( 1 - 2.06e9T + 1.12e20T^{2} \)
17 \( 1 + 1.59e11T + 1.40e22T^{2} \)
19 \( 1 - 3.75e11iT - 1.04e23T^{2} \)
23 \( 1 - 3.51e12iT - 3.24e24T^{2} \)
29 \( 1 - 1.15e12T + 2.10e26T^{2} \)
31 \( 1 + 2.94e13iT - 6.99e26T^{2} \)
37 \( 1 - 1.43e14T + 1.68e28T^{2} \)
41 \( 1 + 3.41e14T + 1.07e29T^{2} \)
43 \( 1 - 1.16e14iT - 2.52e29T^{2} \)
47 \( 1 - 1.28e15iT - 1.25e30T^{2} \)
53 \( 1 + 9.01e14T + 1.08e31T^{2} \)
59 \( 1 + 1.35e16iT - 7.50e31T^{2} \)
61 \( 1 - 1.92e15T + 1.36e32T^{2} \)
67 \( 1 - 3.28e16iT - 7.40e32T^{2} \)
71 \( 1 - 1.05e16iT - 2.10e33T^{2} \)
73 \( 1 + 5.27e16T + 3.46e33T^{2} \)
79 \( 1 + 2.46e16iT - 1.43e34T^{2} \)
83 \( 1 + 1.57e17iT - 3.49e34T^{2} \)
89 \( 1 + 1.76e17T + 1.22e35T^{2} \)
97 \( 1 - 8.50e16T + 5.77e35T^{2} \)
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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

−20.34571198220967397692843563119, −19.00190758990131536135033455742, −17.72161780219575275169908398381, −15.54618515575670601694238318600, −13.02330194840678990713526303929, −11.66607022821518641598049061629, −9.692240712770639587396648781700, −7.65251093993026270030348420835, −4.14517795804589275651047091905, −1.79520958841218477442810051099, 0.49163872616787734541886495472, 4.37445666083121237427271949221, 6.81331802226156844595576856278, 8.626510779856763090465269347117, 10.71803566192251030818862201420, 13.49216942491982570938235538336, 15.39421368100598034060522978462, 16.48525854322644381185407175302, 18.32617090184924152968053851625, 19.74051055015451642268897876038

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