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Properties

Label 2-2e9-16.5-c3-0-9
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 + (−7 − 7i)5-s + 8.94i·7-s − 13.0i·9-s + (40.2 + 40.2i)11-s + (−45 + 45i)13-s − 62.6·15-s − 16·17-s + (−76.0 + 76.0i)19-s + (40.0 + 40.0i)21-s − 8.94i·23-s − 27i·25-s + (62.6 + 62.6i)27-s + (−67 + 67i)29-s − 304.·31-s + ⋯
L(s)  = 1  + (0.860 − 0.860i)3-s + (−0.626 − 0.626i)5-s + 0.482i·7-s − 0.481i·9-s + (1.10 + 1.10i)11-s + (−0.960 + 0.960i)13-s − 1.07·15-s − 0.228·17-s + (−0.917 + 0.917i)19-s + (0.415 + 0.415i)21-s − 0.0810i·23-s − 0.215i·25-s + (0.446 + 0.446i)27-s + (−0.429 + 0.429i)29-s − 1.76·31-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: $\chi_{512} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :3/2),\ 0.382 - 0.923i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.517216393\)
\(L(\frac12)\) \(\approx\) \(1.517216393\)
\(L(\frac{5}{2})\) 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 \( 1 + (7 + 7i)T + 125iT^{2} \)
7 \( 1 - 8.94iT - 343T^{2} \)
11 \( 1 + (-40.2 - 40.2i)T + 1.33e3iT^{2} \)
13 \( 1 + (45 - 45i)T - 2.19e3iT^{2} \)
17 \( 1 + 16T + 4.91e3T^{2} \)
19 \( 1 + (76.0 - 76.0i)T - 6.85e3iT^{2} \)
23 \( 1 + 8.94iT - 1.21e4T^{2} \)
29 \( 1 + (67 - 67i)T - 2.43e4iT^{2} \)
31 \( 1 + 304.T + 2.97e4T^{2} \)
37 \( 1 + (9 + 9i)T + 5.06e4iT^{2} \)
41 \( 1 - 328iT - 6.89e4T^{2} \)
43 \( 1 + (-174. - 174. i)T + 7.95e4iT^{2} \)
47 \( 1 - 411.T + 1.03e5T^{2} \)
53 \( 1 + (-407 - 407i)T + 1.48e5iT^{2} \)
59 \( 1 + (523. + 523. i)T + 2.05e5iT^{2} \)
61 \( 1 + (589 - 589i)T - 2.26e5iT^{2} \)
67 \( 1 + (-210. + 210. i)T - 3.00e5iT^{2} \)
71 \( 1 + 688. iT - 3.57e5T^{2} \)
73 \( 1 + 782iT - 3.89e5T^{2} \)
79 \( 1 - 429.T + 4.93e5T^{2} \)
83 \( 1 + (-22.3 + 22.3i)T - 5.71e5iT^{2} \)
89 \( 1 - 510iT - 7.04e5T^{2} \)
97 \( 1 - 1.21e3T + 9.12e5T^{2} \)
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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

−10.67092341257268014904180995560, −9.179763728242032177867749853834, −9.068677225878005671931261731990, −7.83778286474576325878725049759, −7.27785193714597416890670234329, −6.29120456380256678174106337905, −4.74422320261319637981188967349, −3.93032259580527610682761549003, −2.31664770560685355047153664121, −1.55237870453555415733387352663, 0.40621589848088017111930718689, 2.55685516574066886675947304813, 3.61810407828075536878244882490, 4.09054860516159497268936359523, 5.54461639505563337640437216631, 6.87881027281081264597880881750, 7.63066947726003239474294864112, 8.789060974060028649854387715322, 9.235030646216774357522232324071, 10.44595507994162638112844150984

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