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Properties

Label 2-2e9-32.29-c1-0-3
Degree $2$
Conductor $512$
Sign $0.195 - 0.980i$
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  + (1.70 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.195 - 0.980i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.195 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54206 + 1.26553i\)
\(L(\frac12)\) \(\approx\) \(1.54206 + 1.26553i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.51T + 97T^{2} \)
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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

−10.90816245156844377286552470708, −10.07724022607232699350269828486, −9.330682402827643648654025168143, −8.672444134509992493865069454206, −7.38420758661995690699161603558, −6.60508396574732304213570225049, −5.64845363974742326484798137916, −4.04639799900959809213827091605, −2.99972411211255227672338869200, −2.33073880005159011949646015232, 1.15968335931606288035699992566, 2.48547718427633467974134033716, 3.79358978924234690602699085188, 5.07199755704602448152340340110, 5.94169811290984680695497216588, 7.39382802977332652849392586754, 8.047305660178889925435735199499, 8.989831510141680970451687275536, 9.513943783629391714652731477458, 10.46143794344611691606409036569

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