↑

Properties

Label 1-5225-5225.1659-r0-0-0
Degree $1$
Conductor $5225$
Sign $0.608 + 0.793i$
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  + (0.882 + 0.469i)2-s + (0.374 − 0.927i)3-s + (0.559 + 0.829i)4-s + (0.766 − 0.642i)6-s + (−0.669 + 0.743i)7-s + (0.104 + 0.994i)8-s + (−0.719 − 0.694i)9-s + (0.978 − 0.207i)12-s + (0.241 − 0.970i)13-s + (−0.939 + 0.342i)14-s + (−0.374 + 0.927i)16-s + (−0.173 + 0.984i)17-s + (−0.309 − 0.951i)18-s + (0.438 + 0.898i)21-s + (−0.0348 + 0.999i)23-s + (0.961 + 0.275i)24-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)2-s + (0.374 − 0.927i)3-s + (0.559 + 0.829i)4-s + (0.766 − 0.642i)6-s + (−0.669 + 0.743i)7-s + (0.104 + 0.994i)8-s + (−0.719 − 0.694i)9-s + (0.978 − 0.207i)12-s + (0.241 − 0.970i)13-s + (−0.939 + 0.342i)14-s + (−0.374 + 0.927i)16-s + (−0.173 + 0.984i)17-s + (−0.309 − 0.951i)18-s + (0.438 + 0.898i)21-s + (−0.0348 + 0.999i)23-s + (0.961 + 0.275i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(24.2647\)
Root analytic conductor: \(24.2647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5225} (1659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5225,\ (0:\ ),\ 0.608 + 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.757744108 + 1.361573935i\)
\(L(\frac12)\) \(\approx\) \(2.757744108 + 1.361573935i\)
\(L(1)\) \(\approx\) \(1.777314824 + 0.3047644372i\)
\(L(1)\) \(\approx\) \(1.777314824 + 0.3047644372i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.882 + 0.469i)T \)
3 \( 1 + (0.374 - 0.927i)T \)
7 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.241 - 0.970i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (-0.0348 + 0.999i)T \)
29 \( 1 + (0.438 - 0.898i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.241 - 0.970i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.882 - 0.469i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (0.559 + 0.829i)T \)
67 \( 1 + (-0.438 + 0.898i)T \)
71 \( 1 + (0.0348 + 0.999i)T \)
73 \( 1 + (0.997 + 0.0697i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.848 + 0.529i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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

−17.94052428997318204443008377806, −16.83345396853893459257976449435, −16.23608140566370624310057399928, −15.98509915867882804180452521847, −15.131713855521037293147312587603, −14.36838581778102353233149056152, −13.86587421700685983853846275474, −13.52180775187409263037018050306, −12.50291897561154752227742079080, −11.911745074455008034565698301379, −11.02131179767427782676705775364, −10.595955170538205543470509113006, −9.9398395287049886504904371582, −9.28251797967460287369772929935, −8.66590403256064722828398190396, −7.477105197543025251039885439143, −6.746001653903876731736606674220, −6.18002372192790946551393983996, −5.10296228993957110983994936863, −4.58409418725251753434264385664, −4.00874721747253994356762025990, −3.21028280814189249739987327588, −2.730831407562828335401369754680, −1.75305128750668088327791178347, −0.61075378791450361643232054639, 0.926387059586403431289678986592, 2.22687613372049285103099566295, 2.49119513697432109554144829991, 3.55191245406318531172332218811, 3.88564040818515730598986237190, 5.32875466185994430439339396737, 5.73773877714771207348770655238, 6.34981534958229443022188138701, 7.036458350065984757051700073636, 7.79056266660411862966816727631, 8.39593887880989163465850959673, 8.94857570734662204869241715734, 9.979303153399374962540141558454, 10.89488125379983951820967445038, 11.81381381599696906668801486754, 12.2509126645223983699399514742, 12.83504688337647497105021209473, 13.4625791177616284994844828522, 13.84845370796101721173741611953, 14.83464094377590938679012763539, 15.42710685921508172569576621343, 15.65134745392936190550672106144, 16.76913088152535969224393959538, 17.51614341204578902047687076433, 17.78903367129022978577185438181

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