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Properties

Label 1-5225-5225.3766-r0-0-0
Degree $1$
Conductor $5225$
Sign $-0.828 + 0.560i$
Analytic cond. $24.2647$
Root an. cond. $24.2647$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.997 − 0.0697i)2-s + (−0.939 + 0.342i)3-s + (0.990 + 0.139i)4-s + (0.961 − 0.275i)6-s + (0.913 + 0.406i)7-s + (−0.978 − 0.207i)8-s + (0.766 − 0.642i)9-s + (−0.978 + 0.207i)12-s + (−0.939 − 0.342i)13-s + (−0.882 − 0.469i)14-s + (0.961 + 0.275i)16-s + (−0.997 − 0.0697i)17-s + (−0.809 + 0.587i)18-s + (−0.997 − 0.0697i)21-s + (−0.882 + 0.469i)23-s + (0.990 − 0.139i)24-s + ⋯
L(s)  = 1  + (−0.997 − 0.0697i)2-s + (−0.939 + 0.342i)3-s + (0.990 + 0.139i)4-s + (0.961 − 0.275i)6-s + (0.913 + 0.406i)7-s + (−0.978 − 0.207i)8-s + (0.766 − 0.642i)9-s + (−0.978 + 0.207i)12-s + (−0.939 − 0.342i)13-s + (−0.882 − 0.469i)14-s + (0.961 + 0.275i)16-s + (−0.997 − 0.0697i)17-s + (−0.809 + 0.587i)18-s + (−0.997 − 0.0697i)21-s + (−0.882 + 0.469i)23-s + (0.990 − 0.139i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $-0.828 + 0.560i$
Analytic conductor: \(24.2647\)
Root analytic conductor: \(24.2647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5225} (3766, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5225,\ (0:\ ),\ -0.828 + 0.560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07205815797 + 0.2351715187i\)
\(L(\frac12)\) \(\approx\) \(0.07205815797 + 0.2351715187i\)
\(L(1)\) \(\approx\) \(0.4785558961 + 0.05593955154i\)
\(L(1)\) \(\approx\) \(0.4785558961 + 0.05593955154i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.997 - 0.0697i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.997 - 0.0697i)T \)
23 \( 1 + (-0.882 + 0.469i)T \)
29 \( 1 + (-0.374 - 0.927i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.559 - 0.829i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (0.438 - 0.898i)T \)
59 \( 1 + (-0.374 + 0.927i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.997 + 0.0697i)T \)
71 \( 1 + (0.990 - 0.139i)T \)
73 \( 1 + (0.961 + 0.275i)T \)
79 \( 1 + (0.961 + 0.275i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (0.0348 - 0.999i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.712409994602138030342672973018, −17.21123599093981397838761898085, −16.48361150357622126355262828731, −16.1330095061249821526535636237, −15.13447155818095367752548955944, −14.550644257374996604304542210542, −13.77971614807336130995714684920, −12.67138579969822485314752257975, −12.27830641072618974897799914231, −11.39544433011917211838564892073, −10.958259414513574560188762737020, −10.52914873792510990310140229534, −9.560719171023097905066843921051, −9.02667734021320940341622401524, −7.82064922829626989486631469053, −7.71429478487819247419278779824, −6.81607619275312115642448263445, −6.27393744901645575700061535771, −5.38613868933527083769007132910, −4.700604069125661624127455027732, −3.90113694553735556980000660970, −2.43611826696114683002807109404, −1.94547549160965574445569234003, −1.12813032845809317768519894172, −0.13267699688079998985244564882, 0.84699798566072244074238093075, 1.93812589223513482105701827461, 2.38209438857899716232750915644, 3.6725827639571915211022343789, 4.44915584006654324513260891640, 5.39148275424590234509990718796, 5.79993811824692580315537783121, 6.76807838396873988247645831508, 7.391605769992670129561404274113, 8.04833755745323739128942627694, 8.94245831887944297707803772146, 9.47084637752344690639101281895, 10.30338079282577581386957314450, 10.74387850919102624420988509463, 11.5465684325992193937470913143, 11.90490048505582335296107769102, 12.52588911428852377042082323690, 13.503990663922318290090948123992, 14.591482530452691649276172175316, 15.31106442856274424255729088748, 15.54283068603235454871103280739, 16.48752672158656775465082353103, 17.09072242940477884883398222438, 17.5588461301024440622103229129, 18.13731397311613615699500467877

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