Properties

Label 2-2e9-512.117-c1-0-0
Degree $2$
Conductor $512$
Sign $0.167 - 0.985i$
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  + (−1.03 − 0.964i)2-s + (−0.0450 − 1.83i)3-s + (0.137 + 1.99i)4-s + (−0.818 − 2.95i)5-s + (−1.72 + 1.93i)6-s + (−1.77 + 3.76i)7-s + (1.78 − 2.19i)8-s + (−0.362 + 0.0178i)9-s + (−2.00 + 3.84i)10-s + (−0.558 − 0.393i)11-s + (3.65 − 0.342i)12-s + (−6.03 + 0.744i)13-s + (5.46 − 2.17i)14-s + (−5.38 + 1.63i)15-s + (−3.96 + 0.549i)16-s + (−0.525 + 1.73i)17-s + ⋯
L(s)  = 1  + (−0.731 − 0.682i)2-s + (−0.0259 − 1.05i)3-s + (0.0689 + 0.997i)4-s + (−0.365 − 1.32i)5-s + (−0.703 + 0.791i)6-s + (−0.672 + 1.42i)7-s + (0.630 − 0.776i)8-s + (−0.120 + 0.00594i)9-s + (−0.634 + 1.21i)10-s + (−0.168 − 0.118i)11-s + (1.05 − 0.0988i)12-s + (−1.67 + 0.206i)13-s + (1.46 − 0.580i)14-s + (−1.38 + 0.421i)15-s + (−0.990 + 0.137i)16-s + (−0.127 + 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.167 - 0.985i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.167 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00809260 + 0.00683370i\)
\(L(\frac12)\) \(\approx\) \(0.00809260 + 0.00683370i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 + 0.964i)T \)
good3 \( 1 + (0.0450 + 1.83i)T + (-2.99 + 0.147i)T^{2} \)
5 \( 1 + (0.818 + 2.95i)T + (-4.28 + 2.57i)T^{2} \)
7 \( 1 + (1.77 - 3.76i)T + (-4.44 - 5.41i)T^{2} \)
11 \( 1 + (0.558 + 0.393i)T + (3.70 + 10.3i)T^{2} \)
13 \( 1 + (6.03 - 0.744i)T + (12.6 - 3.15i)T^{2} \)
17 \( 1 + (0.525 - 1.73i)T + (-14.1 - 9.44i)T^{2} \)
19 \( 1 + (0.135 - 1.83i)T + (-18.7 - 2.78i)T^{2} \)
23 \( 1 + (5.44 - 4.03i)T + (6.67 - 22.0i)T^{2} \)
29 \( 1 + (-2.87 + 0.645i)T + (26.2 - 12.3i)T^{2} \)
31 \( 1 + (1.14 + 1.70i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (-0.164 - 0.326i)T + (-22.0 + 29.7i)T^{2} \)
41 \( 1 + (4.34 + 2.60i)T + (19.3 + 36.1i)T^{2} \)
43 \( 1 + (-7.32 - 7.68i)T + (-2.10 + 42.9i)T^{2} \)
47 \( 1 + (-3.04 - 2.50i)T + (9.16 + 46.0i)T^{2} \)
53 \( 1 + (9.99 + 2.24i)T + (47.9 + 22.6i)T^{2} \)
59 \( 1 + (-1.21 + 9.85i)T + (-57.2 - 14.3i)T^{2} \)
61 \( 1 + (3.92 + 8.84i)T + (-40.9 + 45.1i)T^{2} \)
67 \( 1 + (3.13 - 8.12i)T + (-49.6 - 44.9i)T^{2} \)
71 \( 1 + (-6.01 - 6.63i)T + (-6.95 + 70.6i)T^{2} \)
73 \( 1 + (-3.47 - 7.35i)T + (-46.3 + 56.4i)T^{2} \)
79 \( 1 + (0.229 - 2.33i)T + (-77.4 - 15.4i)T^{2} \)
83 \( 1 + (6.61 - 13.1i)T + (-49.4 - 66.6i)T^{2} \)
89 \( 1 + (12.8 + 9.52i)T + (25.8 + 85.1i)T^{2} \)
97 \( 1 + (0.569 + 0.113i)T + (89.6 + 37.1i)T^{2} \)
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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

−11.44551759411031256020541102123, −9.783538837125137132090941947871, −9.457726689091873130086061986595, −8.312630229970707082868024163933, −7.897013775374334087729621827270, −6.75263183512521542813402447865, −5.55230927925160910701155165630, −4.27790214903545233754807362215, −2.64750443805021594898646840562, −1.66478112997396871407935083362, 0.00766093201329391945872098568, 2.73205340337234302110114951880, 4.05362550679658438608983438217, 4.93554549500261000235208642894, 6.45481783784768443669285733789, 7.20718894317534490717781748150, 7.64064856172019870332391223721, 9.195581743100665728040757023313, 10.06844492095556256688758908526, 10.37170990189774690343833416505

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