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Properties

Label 2-2e9-16.13-c3-0-40
Degree $2$
Conductor $512$
Sign $0.382 + 0.923i$
Analytic cond. $30.2089$
Root an. cond. $5.49626$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.47 + 4.47i)3-s + (11 − 11i)5-s − 26.8i·7-s + 13.0i·9-s + (−31.3 + 31.3i)11-s + (−23 − 23i)13-s + 98.3·15-s + 96·17-s + (−76.0 − 76.0i)19-s + (120. − 120. i)21-s − 116. i·23-s − 117i·25-s + (62.6 − 62.6i)27-s + (103 + 103i)29-s − 196.·31-s + ⋯
L(s)  = 1  + (0.860 + 0.860i)3-s + (0.983 − 0.983i)5-s − 1.44i·7-s + 0.481i·9-s + (−0.858 + 0.858i)11-s + (−0.490 − 0.490i)13-s + 1.69·15-s + 1.36·17-s + (−0.917 − 0.917i)19-s + (1.24 − 1.24i)21-s − 1.05i·23-s − 0.936i·25-s + (0.446 − 0.446i)27-s + (0.659 + 0.659i)29-s − 1.14·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(30.2089\)
Root analytic conductor: \(5.49626\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :3/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.716595423\)
\(L(\frac12)\) \(\approx\) \(2.716595423\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-4.47 - 4.47i)T + 27iT^{2} \)
5 \( 1 + (-11 + 11i)T - 125iT^{2} \)
7 \( 1 + 26.8iT - 343T^{2} \)
11 \( 1 + (31.3 - 31.3i)T - 1.33e3iT^{2} \)
13 \( 1 + (23 + 23i)T + 2.19e3iT^{2} \)
17 \( 1 - 96T + 4.91e3T^{2} \)
19 \( 1 + (76.0 + 76.0i)T + 6.85e3iT^{2} \)
23 \( 1 + 116. iT - 1.21e4T^{2} \)
29 \( 1 + (-103 - 103i)T + 2.43e4iT^{2} \)
31 \( 1 + 196.T + 2.97e4T^{2} \)
37 \( 1 + (75 - 75i)T - 5.06e4iT^{2} \)
41 \( 1 + 312iT - 6.89e4T^{2} \)
43 \( 1 + (254. - 254. i)T - 7.95e4iT^{2} \)
47 \( 1 - 89.4T + 1.03e5T^{2} \)
53 \( 1 + (-325 + 325i)T - 1.48e5iT^{2} \)
59 \( 1 + (-263. + 263. i)T - 2.05e5iT^{2} \)
61 \( 1 + (39 + 39i)T + 2.26e5iT^{2} \)
67 \( 1 + (-424. - 424. i)T + 3.00e5iT^{2} \)
71 \( 1 + 366. iT - 3.57e5T^{2} \)
73 \( 1 + 1.13e3iT - 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 + (-523. - 523. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.41e3iT - 7.04e5T^{2} \)
97 \( 1 - 464T + 9.12e5T^{2} \)
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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

−10.24442330767183394013404556213, −9.564787015741027754419245969779, −8.687067317828155181794012821808, −7.80212267618262998416236866593, −6.76777146624414154957104501040, −5.19472193075313141018444449403, −4.65607823787197854691941357074, −3.52742088288099054857736747522, −2.23197568879232002540408932477, −0.70967894729695865639604611529, 1.79266415347160940507018781201, 2.49245756238321598431887803574, 3.27806816098684393484094731038, 5.47938608427385595840416276788, 5.95120518406224164900918631574, 7.08905169218016160761473533387, 8.018721490448066595505239326500, 8.724746283267947180562458096144, 9.734103403898915333909795689692, 10.48490730288237571644710801565

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