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Properties

Label 2-2e9-64.13-c1-0-3
Degree $2$
Conductor $512$
Sign $0.310 - 0.950i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.31 − 0.876i)3-s + (−3.52 + 0.700i)5-s + (1.02 + 2.47i)7-s + (−0.196 + 0.473i)9-s + (−1.67 + 2.51i)11-s + (4.41 + 0.878i)13-s + (−4.00 + 4.00i)15-s + (1.12 + 1.12i)17-s + (−0.432 + 2.17i)19-s + (3.51 + 2.34i)21-s + (−4.52 − 1.87i)23-s + (7.29 − 3.02i)25-s + (1.08 + 5.43i)27-s + (3.43 + 5.14i)29-s + 2.88i·31-s + ⋯
L(s)  = 1  + (0.757 − 0.505i)3-s + (−1.57 + 0.313i)5-s + (0.387 + 0.935i)7-s + (−0.0653 + 0.157i)9-s + (−0.506 + 0.757i)11-s + (1.22 + 0.243i)13-s + (−1.03 + 1.03i)15-s + (0.273 + 0.273i)17-s + (−0.0993 + 0.499i)19-s + (0.766 + 0.512i)21-s + (−0.943 − 0.390i)23-s + (1.45 − 0.604i)25-s + (0.207 + 1.04i)27-s + (0.637 + 0.954i)29-s + 0.518i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.310 - 0.950i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01688 + 0.737785i\)
\(L(\frac12)\) \(\approx\) \(1.01688 + 0.737785i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 + 0.876i)T + (1.14 - 2.77i)T^{2} \)
5 \( 1 + (3.52 - 0.700i)T + (4.61 - 1.91i)T^{2} \)
7 \( 1 + (-1.02 - 2.47i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.67 - 2.51i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + (-4.41 - 0.878i)T + (12.0 + 4.97i)T^{2} \)
17 \( 1 + (-1.12 - 1.12i)T + 17iT^{2} \)
19 \( 1 + (0.432 - 2.17i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (4.52 + 1.87i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.43 - 5.14i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 - 2.88iT - 31T^{2} \)
37 \( 1 + (1.20 + 6.05i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (3.20 + 1.32i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.16 - 1.44i)T + (16.4 + 39.7i)T^{2} \)
47 \( 1 + (2.37 + 2.37i)T + 47iT^{2} \)
53 \( 1 + (-3.20 + 4.78i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (1.01 - 0.201i)T + (54.5 - 22.5i)T^{2} \)
61 \( 1 + (-2.23 + 1.49i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (12.1 - 8.10i)T + (25.6 - 61.8i)T^{2} \)
71 \( 1 + (-4.63 - 11.1i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-3.99 + 9.65i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.05 + 1.05i)T - 79iT^{2} \)
83 \( 1 + (-1.67 + 8.41i)T + (-76.6 - 31.7i)T^{2} \)
89 \( 1 + (5.40 - 2.23i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−11.14397340002831319511778122828, −10.36970597977075797503570204710, −8.804622571861710229750939283794, −8.334213869012239042121357847254, −7.70699371923024702675786891280, −6.79144277231302861424136847881, −5.41912665359476063892040889633, −4.15100938047049507541551319000, −3.14233034741144154901187070820, −1.90148878387865252629123381822, 0.70953496467587780144780183780, 3.12803992561712809189366631513, 3.85751880122281552479575119004, 4.57538198243508336703005399622, 6.11062352701556493783456546447, 7.47629267623699660557517359566, 8.194406393129611610896228095687, 8.596299372242581593682462127085, 9.828704401760301736897831578185, 10.83790388152355498260527804420

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