Properties

Label 2-2e9-512.109-c1-0-3
Degree $2$
Conductor $512$
Sign $-0.658 + 0.752i$
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  + (0.437 + 1.34i)2-s + (0.0973 − 0.789i)3-s + (−1.61 + 1.17i)4-s + (−1.66 + 0.373i)5-s + (1.10 − 0.213i)6-s + (−1.54 + 1.14i)7-s + (−2.28 − 1.66i)8-s + (2.29 + 0.575i)9-s + (−1.22 − 2.07i)10-s + (−2.11 − 0.155i)11-s + (0.770 + 1.39i)12-s + (−5.70 − 4.01i)13-s + (−2.21 − 1.57i)14-s + (0.132 + 1.34i)15-s + (1.23 − 3.80i)16-s + (−0.879 − 0.0866i)17-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.0561 − 0.455i)3-s + (−0.809 + 0.587i)4-s + (−0.743 + 0.166i)5-s + (0.450 − 0.0873i)6-s + (−0.583 + 0.432i)7-s + (−0.809 − 0.587i)8-s + (0.765 + 0.191i)9-s + (−0.388 − 0.655i)10-s + (−0.637 − 0.0470i)11-s + (0.222 + 0.401i)12-s + (−1.58 − 1.11i)13-s + (−0.591 − 0.421i)14-s + (0.0342 + 0.347i)15-s + (0.309 − 0.951i)16-s + (−0.213 − 0.0210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0549881 - 0.121228i\)
\(L(\frac12)\) \(\approx\) \(0.0549881 - 0.121228i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.437 - 1.34i)T \)
good3 \( 1 + (-0.0973 + 0.789i)T + (-2.91 - 0.728i)T^{2} \)
5 \( 1 + (1.66 - 0.373i)T + (4.51 - 2.13i)T^{2} \)
7 \( 1 + (1.54 - 1.14i)T + (2.03 - 6.69i)T^{2} \)
11 \( 1 + (2.11 + 0.155i)T + (10.8 + 1.61i)T^{2} \)
13 \( 1 + (5.70 + 4.01i)T + (4.37 + 12.2i)T^{2} \)
17 \( 1 + (0.879 + 0.0866i)T + (16.6 + 3.31i)T^{2} \)
19 \( 1 + (-1.03 - 2.67i)T + (-14.0 + 12.7i)T^{2} \)
23 \( 1 + (4.76 + 0.234i)T + (22.8 + 2.25i)T^{2} \)
29 \( 1 + (1.64 + 3.27i)T + (-17.2 + 23.2i)T^{2} \)
31 \( 1 + (-1.04 - 5.25i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (5.30 + 5.05i)T + (1.81 + 36.9i)T^{2} \)
41 \( 1 + (-0.919 - 0.434i)T + (26.0 + 31.6i)T^{2} \)
43 \( 1 + (3.05 - 3.91i)T + (-10.4 - 41.7i)T^{2} \)
47 \( 1 + (-1.77 + 0.538i)T + (39.0 - 26.1i)T^{2} \)
53 \( 1 + (-0.0417 + 0.0829i)T + (-31.5 - 42.5i)T^{2} \)
59 \( 1 + (-7.07 - 10.0i)T + (-19.8 + 55.5i)T^{2} \)
61 \( 1 + (-2.40 - 1.36i)T + (31.3 + 52.3i)T^{2} \)
67 \( 1 + (3.10 - 0.858i)T + (57.4 - 34.4i)T^{2} \)
71 \( 1 + (-1.79 + 3.00i)T + (-33.4 - 62.6i)T^{2} \)
73 \( 1 + (-1.20 - 0.893i)T + (21.1 + 69.8i)T^{2} \)
79 \( 1 + (-1.78 - 3.33i)T + (-43.8 + 65.6i)T^{2} \)
83 \( 1 + (8.62 - 8.21i)T + (4.07 - 82.9i)T^{2} \)
89 \( 1 + (8.59 - 0.422i)T + (88.5 - 8.72i)T^{2} \)
97 \( 1 + (-8.03 + 12.0i)T + (-37.1 - 89.6i)T^{2} \)
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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

−11.89801695407044108468015215913, −10.28836993143822782468493451989, −9.694083099619259408300401492318, −8.355584953992332924350837609699, −7.62103815499749613738670041294, −7.17619740444067125956963249703, −5.96114387493484591287430003862, −5.04534901223121675258058947599, −3.88415318197021548964106829829, −2.66009041467776795468573760020, 0.06715338839042092607397195771, 2.15630653379038320312745614305, 3.56039556876699173873522281544, 4.35211268766371442747867304849, 5.09145827010554655737321284471, 6.69710400725311493443648507624, 7.63055092732649954886364688538, 8.892044600242460645451152685575, 9.890488996638373618839363473675, 10.08700301735702180645755540827

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