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Properties

Label 2-2e9-16.13-c1-0-12
Degree $2$
Conductor $512$
Sign $-0.707 + 0.707i$
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  + (−1.84 − 1.84i)3-s + (2.41 − 2.41i)5-s − 1.53i·7-s + 3.82i·9-s + (3.37 − 3.37i)11-s + (0.414 + 0.414i)13-s − 8.92·15-s + 2.82·17-s + (−0.317 − 0.317i)19-s + (−2.82 + 2.82i)21-s + 5.86i·23-s − 6.65i·25-s + (1.53 − 1.53i)27-s + (−3.24 − 3.24i)29-s − 7.39·31-s + ⋯
L(s)  = 1  + (−1.06 − 1.06i)3-s + (1.07 − 1.07i)5-s − 0.578i·7-s + 1.27i·9-s + (1.01 − 1.01i)11-s + (0.114 + 0.114i)13-s − 2.30·15-s + 0.685·17-s + (−0.0727 − 0.0727i)19-s + (−0.617 + 0.617i)21-s + 1.22i·23-s − 1.33i·25-s + (0.294 − 0.294i)27-s + (−0.602 − 0.602i)29-s − 1.32·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.462062 - 1.11551i\)
\(L(\frac12)\) \(\approx\) \(0.462062 - 1.11551i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.84 + 1.84i)T + 3iT^{2} \)
5 \( 1 + (-2.41 + 2.41i)T - 5iT^{2} \)
7 \( 1 + 1.53iT - 7T^{2} \)
11 \( 1 + (-3.37 + 3.37i)T - 11iT^{2} \)
13 \( 1 + (-0.414 - 0.414i)T + 13iT^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + (0.317 + 0.317i)T + 19iT^{2} \)
23 \( 1 - 5.86iT - 23T^{2} \)
29 \( 1 + (3.24 + 3.24i)T + 29iT^{2} \)
31 \( 1 + 7.39T + 31T^{2} \)
37 \( 1 + (3.58 - 3.58i)T - 37iT^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + (-1.84 + 1.84i)T - 43iT^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 + (5.24 - 5.24i)T - 53iT^{2} \)
59 \( 1 + (-1.84 + 1.84i)T - 59iT^{2} \)
61 \( 1 + (9.24 + 9.24i)T + 61iT^{2} \)
67 \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \)
71 \( 1 - 11.9iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 - 6.12T + 79T^{2} \)
83 \( 1 + (-2.48 - 2.48i)T + 83iT^{2} \)
89 \( 1 - 0.828iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−10.78023239001663579809862221954, −9.590139701370607431367062884735, −8.879571920160829405405846710909, −7.67694796525038707173714221835, −6.73200675288096731388004164441, −5.76530334127392345837693184033, −5.41385865811822707890295276021, −3.84152083693077467313520762439, −1.69610534499010123465804651596, −0.887355469493061669381303777417, 2.04500901582175209608292805278, 3.53704733589736413643395014094, 4.75178476715400533185907338332, 5.73887640354888048842474726919, 6.32950399028875257059212548573, 7.29507387621874326483291651542, 9.084657616330405869426171239068, 9.611829668066953224921363763148, 10.49882949504746278096282904340, 10.87092260250064545016117406998

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