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Properties

Label 2-2e9-64.13-c1-0-7
Degree $2$
Conductor $512$
Sign $0.999 + 0.00795i$
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  + (1.06 − 0.714i)3-s + (0.330 − 0.0657i)5-s + (0.739 + 1.78i)7-s + (−0.515 + 1.24i)9-s + (0.971 − 1.45i)11-s + (3.70 + 0.737i)13-s + (0.306 − 0.306i)15-s + (4.47 + 4.47i)17-s + (1.16 − 5.83i)19-s + (2.06 + 1.38i)21-s + (1.28 + 0.531i)23-s + (−4.51 + 1.86i)25-s + (1.09 + 5.48i)27-s + (−3.04 − 4.56i)29-s − 10.2i·31-s + ⋯
L(s)  = 1  + (0.617 − 0.412i)3-s + (0.147 − 0.0294i)5-s + (0.279 + 0.674i)7-s + (−0.171 + 0.414i)9-s + (0.292 − 0.438i)11-s + (1.02 + 0.204i)13-s + (0.0791 − 0.0791i)15-s + (1.08 + 1.08i)17-s + (0.266 − 1.33i)19-s + (0.450 + 0.301i)21-s + (0.267 + 0.110i)23-s + (−0.902 + 0.373i)25-s + (0.209 + 1.05i)27-s + (−0.566 − 0.847i)29-s − 1.84i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.999 + 0.00795i$
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: \((2,\ 512,\ (\ :1/2),\ 0.999 + 0.00795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90933 - 0.00759675i\)
\(L(\frac12)\) \(\approx\) \(1.90933 - 0.00759675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 + 0.714i)T + (1.14 - 2.77i)T^{2} \)
5 \( 1 + (-0.330 + 0.0657i)T + (4.61 - 1.91i)T^{2} \)
7 \( 1 + (-0.739 - 1.78i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.971 + 1.45i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + (-3.70 - 0.737i)T + (12.0 + 4.97i)T^{2} \)
17 \( 1 + (-4.47 - 4.47i)T + 17iT^{2} \)
19 \( 1 + (-1.16 + 5.83i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-1.28 - 0.531i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (3.04 + 4.56i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + 10.2iT - 31T^{2} \)
37 \( 1 + (-0.910 - 4.57i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-2.66 - 1.10i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (5.83 + 3.89i)T + (16.4 + 39.7i)T^{2} \)
47 \( 1 + (-0.0482 - 0.0482i)T + 47iT^{2} \)
53 \( 1 + (6.43 - 9.62i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (2.89 - 0.576i)T + (54.5 - 22.5i)T^{2} \)
61 \( 1 + (-0.675 + 0.451i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (2.41 - 1.61i)T + (25.6 - 61.8i)T^{2} \)
71 \( 1 + (-2.88 - 6.96i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-1.92 + 4.64i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (10.5 - 10.5i)T - 79iT^{2} \)
83 \( 1 + (0.0104 - 0.0525i)T + (-76.6 - 31.7i)T^{2} \)
89 \( 1 + (7.52 - 3.11i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 12.1iT - 97T^{2} \)
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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

−11.15649472781286899204900917969, −9.851555656901666634807801125678, −8.938800643028640137237302883763, −8.253062479187428346853695007293, −7.51560319238632245249869719137, −6.16250859867334785316503343237, −5.46952218570389283916541770768, −3.97980807215250529456071057712, −2.77361740800544328798532871356, −1.57340982282797598238798562178, 1.37741089355922876925431736517, 3.20986055506873280261733952436, 3.88358984721187289978758936617, 5.18642433050315915112960376844, 6.30210967483022341968149824604, 7.42301342868704525010738086077, 8.251618144978911547931549641752, 9.212791012923248544368960408202, 9.937820905062324600971252744381, 10.71606781454281746899399874725

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