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Properties

Label 2-2e9-32.21-c1-0-9
Degree $2$
Conductor $512$
Sign $0.493 + 0.870i$
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  + (2.60 − 1.07i)3-s + (0.292 − 0.707i)5-s + (−1.68 − 1.68i)7-s + (3.50 − 3.50i)9-s + (0.808 + 0.334i)11-s + (0.451 + 1.09i)13-s − 2.15i·15-s − 0.224i·17-s + (−1.19 − 2.87i)19-s + (−6.21 − 2.57i)21-s + (3.68 − 3.68i)23-s + (3.12 + 3.12i)25-s + (2.11 − 5.09i)27-s + (−5.66 + 2.34i)29-s + 6.82·31-s + ⋯
L(s)  = 1  + (1.50 − 0.623i)3-s + (0.130 − 0.316i)5-s + (−0.637 − 0.637i)7-s + (1.16 − 1.16i)9-s + (0.243 + 0.100i)11-s + (0.125 + 0.302i)13-s − 0.557i·15-s − 0.0545i·17-s + (−0.273 − 0.660i)19-s + (−1.35 − 0.561i)21-s + (0.768 − 0.768i)23-s + (0.624 + 0.624i)25-s + (0.406 − 0.981i)27-s + (−1.05 + 0.435i)29-s + 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.493 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.493 + 0.870i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.493 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94300 - 1.13222i\)
\(L(\frac12)\) \(\approx\) \(1.94300 - 1.13222i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 + 1.07i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.292 + 0.707i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.68 + 1.68i)T + 7iT^{2} \)
11 \( 1 + (-0.808 - 0.334i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.451 - 1.09i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.224iT - 17T^{2} \)
19 \( 1 + (1.19 + 2.87i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.68 + 3.68i)T - 23iT^{2} \)
29 \( 1 + (5.66 - 2.34i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + (4.09 - 9.87i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (6.37 - 6.37i)T - 41iT^{2} \)
43 \( 1 + (4.60 + 1.90i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 0.542iT - 47T^{2} \)
53 \( 1 + (-9.46 - 3.91i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.39 - 3.36i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.962 - 0.398i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (3.57 - 1.48i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-5.39 - 5.39i)T + 71iT^{2} \)
73 \( 1 + (-5.15 + 5.15i)T - 73iT^{2} \)
79 \( 1 + 8.39iT - 79T^{2} \)
83 \( 1 + (-4.64 - 11.2i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-5.92 - 5.92i)T + 89iT^{2} \)
97 \( 1 + 4.19T + 97T^{2} \)
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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

−10.53500091487969579834553593402, −9.601296436351518620798511085747, −8.867688008734703820173292757751, −8.223270799838756898813286872717, −7.03793210563719365306550255668, −6.65337519290527160229017220695, −4.86225649528618927880186561605, −3.63518705261291005567348330064, −2.73478246322379693595472037234, −1.32198757493050711272928574702, 2.16234318048058964640696623561, 3.16991187470117014546410056357, 3.92625749656851438211193264133, 5.35631907744812225186760133080, 6.54217200234960753222745376722, 7.67230761765633295181119874136, 8.593494912423834695372415150884, 9.198710829287059732732654671350, 9.962104898183730627410382327772, 10.72150754185838530107858924022

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