↑

Properties

Label 1-5225-5225.2759-r0-0-0
Degree $1$
Conductor $5225$
Sign $0.932 + 0.361i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.848 + 0.529i)2-s + (0.615 + 0.788i)3-s + (0.438 − 0.898i)4-s + (−0.939 − 0.342i)6-s + (−0.669 + 0.743i)7-s + (0.104 + 0.994i)8-s + (−0.241 + 0.970i)9-s + (0.978 − 0.207i)12-s + (−0.961 + 0.275i)13-s + (0.173 − 0.984i)14-s + (−0.615 − 0.788i)16-s + (−0.766 − 0.642i)17-s + (−0.309 − 0.951i)18-s + (−0.997 − 0.0697i)21-s + (0.882 − 0.469i)23-s + (−0.719 + 0.694i)24-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (0.615 + 0.788i)3-s + (0.438 − 0.898i)4-s + (−0.939 − 0.342i)6-s + (−0.669 + 0.743i)7-s + (0.104 + 0.994i)8-s + (−0.241 + 0.970i)9-s + (0.978 − 0.207i)12-s + (−0.961 + 0.275i)13-s + (0.173 − 0.984i)14-s + (−0.615 − 0.788i)16-s + (−0.766 − 0.642i)17-s + (−0.309 − 0.951i)18-s + (−0.997 − 0.0697i)21-s + (0.882 − 0.469i)23-s + (−0.719 + 0.694i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $0.932 + 0.361i$
Analytic conductor: \(24.2647\)
Root analytic conductor: \(24.2647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5225} (2759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5225,\ (0:\ ),\ 0.932 + 0.361i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6891060985 + 0.1288932662i\)
\(L(\frac12)\) \(\approx\) \(0.6891060985 + 0.1288932662i\)
\(L(1)\) \(\approx\) \(0.6054350358 + 0.3304040050i\)
\(L(1)\) \(\approx\) \(0.6054350358 + 0.3304040050i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.848 + 0.529i)T \)
3 \( 1 + (0.615 + 0.788i)T \)
7 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (-0.961 + 0.275i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.882 - 0.469i)T \)
29 \( 1 + (-0.997 + 0.0697i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 - T \)
41 \( 1 + (0.961 + 0.275i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.848 - 0.529i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (0.438 - 0.898i)T \)
67 \( 1 + (0.997 - 0.0697i)T \)
71 \( 1 + (-0.882 - 0.469i)T \)
73 \( 1 + (-0.559 + 0.829i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.0348 - 0.999i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.79110743040293926162320786997, −17.4663305166311186765594513585, −16.95676415199271269084161207363, −16.0303318506211601148820308322, −15.344453554976879945102469293217, −14.604570988005669276013198361, −13.69493285201881264346827569127, −13.10214537319222581765188717886, −12.645691951685463127451989579399, −11.997898493664883497253596857241, −11.14970491559680091775865601857, −10.45067185534090350081947996690, −9.77457562718579996642358278116, −9.12747501976606761805517654771, −8.54186150345777402942626818132, −7.710755802029707930006219879486, −7.12950441252852108913720397470, −6.7630696695889942649717180594, −5.79827834163845319018075271348, −4.50480255681397851395600104501, −3.649618159480985883686433167740, −3.07559068595355695258292930828, −2.322540157107031589501323020841, −1.55571762264956153624683365331, −0.69733532827878879634972154428, 0.3002912037788447090305158983, 1.833428217230845032716732509993, 2.50820185113724555580548573377, 3.06323852161247732825316915716, 4.25621950455486842345624190698, 5.02836199522701456276453191116, 5.5689123216875957516898271112, 6.5963367060559814291625064388, 7.11015056430998651917205672612, 8.01583411018641148339709276921, 8.64745005229082917564978846896, 9.33212671164925473238256840478, 9.63508066127206470079448344983, 10.298454892536918082786989028431, 11.22495722557463805032215321422, 11.6650783194243698837640073555, 12.82533398136182114338336654842, 13.454672510845048084701076746648, 14.52765603948896964375085231125, 14.72156439495048817420806952701, 15.55091986930853904850463468187, 15.93808462951891787003253349892, 16.640225428192630149103902763870, 17.15870686421014761164312371525, 18.023383631310558748958212213793

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