Properties

Label 2-825-11.3-c1-0-16
Degree $2$
Conductor $825$
Sign $0.112 - 0.993i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
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 + 0.749i)2-s + (0.309 + 0.951i)3-s + (−0.115 + 0.354i)4-s + (−1.03 − 0.749i)6-s + (0.810 − 2.49i)7-s + (−0.935 − 2.87i)8-s + (−0.809 + 0.587i)9-s + (−3.05 − 1.30i)11-s − 0.372·12-s + (3.13 − 2.27i)13-s + (1.03 + 3.18i)14-s + (2.52 + 1.83i)16-s + (4.61 + 3.35i)17-s + (0.394 − 1.21i)18-s + (2.58 + 7.94i)19-s + ⋯
L(s)  = 1  + (−0.729 + 0.530i)2-s + (0.178 + 0.549i)3-s + (−0.0575 + 0.177i)4-s + (−0.421 − 0.306i)6-s + (0.306 − 0.942i)7-s + (−0.330 − 1.01i)8-s + (−0.269 + 0.195i)9-s + (−0.919 − 0.392i)11-s − 0.107·12-s + (0.869 − 0.631i)13-s + (0.276 + 0.850i)14-s + (0.630 + 0.457i)16-s + (1.11 + 0.812i)17-s + (0.0929 − 0.285i)18-s + (0.592 + 1.82i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.112 - 0.993i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.112 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.799185 + 0.713590i\)
\(L(\frac12)\) \(\approx\) \(0.799185 + 0.713590i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 \)
11 \( 1 + (3.05 + 1.30i)T \)
good2 \( 1 + (1.03 - 0.749i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-0.810 + 2.49i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.13 + 2.27i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.61 - 3.35i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.58 - 7.94i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.15T + 23T^{2} \)
29 \( 1 + (2.71 - 8.36i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.55 + 4.03i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.13 - 3.47i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.41 + 4.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.75T + 43T^{2} \)
47 \( 1 + (1.71 + 5.28i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.03 - 5.11i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.96 + 6.04i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-11.0 - 8.03i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.241T + 67T^{2} \)
71 \( 1 + (-3.20 - 2.32i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.78 + 8.58i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.60 + 4.79i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.76 - 4.91i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 5.39T + 89T^{2} \)
97 \( 1 + (-11.2 + 8.14i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.40924419834136964419951447730, −9.587675785708237854815806105742, −8.424926785690920726815317643036, −8.042106105365714054740635878294, −7.37345268257574825846148813045, −6.11279485164074240304341885990, −5.22204384178276634416684669607, −3.76577051045802694073081324740, −3.34348182193002673890488823886, −1.10145466815454580139208797296, 0.862196188365969265157326788888, 2.20270622921920968956750620171, 2.95071009187785418261692838801, 4.88134109034875686980513194625, 5.52736456480220145890360147672, 6.65539033262086776632307097400, 7.75609557884888574133154555473, 8.465459284183678396032121374139, 9.287991981766367665157854555348, 9.784875362265851443659234782954

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