↑

Properties

Label 2-2e9-32.21-c1-0-6
Degree $2$
Conductor $512$
Sign $-0.555 + 0.831i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 + 0.707i)3-s + (−1.29 + 3.12i)5-s + (−1 − i)7-s + (0.292 − 0.292i)9-s + (−0.292 − 0.121i)11-s + (0.707 + 1.70i)13-s − 6.24i·15-s − 2.82i·17-s + (−2.29 − 5.53i)19-s + (2.41 + 0.999i)21-s + (−0.171 + 0.171i)23-s + (−4.53 − 4.53i)25-s + (1.82 − 4.41i)27-s + (−2.70 + 1.12i)29-s − 4·31-s + ⋯
L(s)  = 1  + (−0.985 + 0.408i)3-s + (−0.578 + 1.39i)5-s + (−0.377 − 0.377i)7-s + (0.0976 − 0.0976i)9-s + (−0.0883 − 0.0365i)11-s + (0.196 + 0.473i)13-s − 1.61i·15-s − 0.685i·17-s + (−0.526 − 1.26i)19-s + (0.526 + 0.218i)21-s + (−0.0357 + 0.0357i)23-s + (−0.907 − 0.907i)25-s + (0.351 − 0.849i)27-s + (−0.502 + 0.208i)29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-0.555 + 0.831i$
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: \(1\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ -0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 0.707i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.29 - 3.12i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (0.292 + 0.121i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.707 - 1.70i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + (2.29 + 5.53i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.171 - 0.171i)T - 23iT^{2} \)
29 \( 1 + (2.70 - 1.12i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-0.707 + 1.70i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-5.82 + 5.82i)T - 41iT^{2} \)
43 \( 1 + (7.94 + 3.29i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 + (7.53 + 3.12i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.53 - 6.12i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.707 - 0.292i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (3.70 - 1.53i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-0.171 - 0.171i)T + 71iT^{2} \)
73 \( 1 + (7 - 7i)T - 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (-2.53 - 6.12i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-2.65 - 2.65i)T + 89iT^{2} \)
97 \( 1 + 1.51T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.92901620157090905380783494542, −10.01564437319546392149777844753, −8.958215261938739257232329653632, −7.54071892197507665960018387457, −6.87588371295695491893327934543, −6.07768546722516718535178817222, −4.87159611387599575493655641723, −3.79840407367954204567949471673, −2.64383555011288117171709345271, 0, 1.47767808027432794889727811471, 3.56197289972756328074027556091, 4.74682585856384818012356174781, 5.70164067647619500056107864730, 6.32444256351903877454830991580, 7.73638679902001296997842820626, 8.448433349133534694166976525597, 9.336876049357029676101445323102, 10.45108017839888821937804282428

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