↑

Properties

Label 1-55e2-3025.873-r1-0-0
Degree $1$
Conductor $3025$
Sign $0.669 + 0.742i$
Analytic cond. $325.081$
Root an. cond. $325.081$
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  + (−0.989 + 0.142i)2-s + (−0.951 + 0.309i)3-s + (0.959 − 0.281i)4-s + (0.897 − 0.441i)6-s + (0.0570 + 0.998i)7-s + (−0.909 + 0.415i)8-s + (0.809 − 0.587i)9-s + (−0.825 + 0.564i)12-s + (0.791 + 0.610i)13-s + (−0.198 − 0.980i)14-s + (0.841 − 0.540i)16-s + (−0.856 − 0.516i)17-s + (−0.717 + 0.696i)18-s + (0.654 − 0.755i)19-s + (−0.362 − 0.931i)21-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (−0.951 + 0.309i)3-s + (0.959 − 0.281i)4-s + (0.897 − 0.441i)6-s + (0.0570 + 0.998i)7-s + (−0.909 + 0.415i)8-s + (0.809 − 0.587i)9-s + (−0.825 + 0.564i)12-s + (0.791 + 0.610i)13-s + (−0.198 − 0.980i)14-s + (0.841 − 0.540i)16-s + (−0.856 − 0.516i)17-s + (−0.717 + 0.696i)18-s + (0.654 − 0.755i)19-s + (−0.362 − 0.931i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(325.081\)
Root analytic conductor: \(325.081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (1:\ ),\ 0.669 + 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9820094844 + 0.4368115130i\)
\(L(\frac12)\) \(\approx\) \(0.9820094844 + 0.4368115130i\)
\(L(1)\) \(\approx\) \(0.5807428031 + 0.1441648010i\)
\(L(1)\) \(\approx\) \(0.5807428031 + 0.1441648010i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.989 + 0.142i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + (0.0570 + 0.998i)T \)
13 \( 1 + (0.791 + 0.610i)T \)
17 \( 1 + (-0.856 - 0.516i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (0.967 - 0.254i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (0.610 + 0.791i)T \)
37 \( 1 + (0.791 - 0.610i)T \)
41 \( 1 + (-0.985 - 0.170i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (0.170 + 0.985i)T \)
53 \( 1 + (0.633 + 0.774i)T \)
59 \( 1 + (-0.696 + 0.717i)T \)
61 \( 1 + (0.696 - 0.717i)T \)
67 \( 1 + (0.717 - 0.696i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (-0.633 + 0.774i)T \)
79 \( 1 + (0.736 + 0.676i)T \)
83 \( 1 + (-0.967 - 0.254i)T \)
89 \( 1 + (0.921 - 0.389i)T \)
97 \( 1 + (-0.113 - 0.993i)T \)
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

−18.63755887844204263266911277620, −17.990868225844682675674872571361, −17.4596412644162342318105611345, −16.81814904392521026958527561570, −16.32543822570065697961600468273, −15.56076011314583951852362361794, −14.84458101956456434410775713098, −13.475012296081192953654862621392, −13.18036869291700365654825715558, −12.208227267977815940187290267714, −11.443223216319859291046469736073, −10.96880560256100382953197950340, −10.23834347191033447901800992942, −9.84750894612066273288500128838, −8.5640766198317787222626872316, −8.05919412987559732423970707860, −7.13956542497557260747263364105, −6.68811856500476990511459141015, −5.91468905003527222306668117289, −5.00773291925292684489346906927, −3.94967537512481606678157087935, −3.15217920101986826964500999062, −1.87754697275856456187850697308, −1.12927683038556212380463865283, −0.54053683444986171000826230314, 0.55416003875155715895131853869, 1.353709709985337273498399728136, 2.39100040216601110770189487878, 3.2229820753417859790538469345, 4.56846854988641051360259067400, 5.18812460659052297416374022092, 6.14816556299348559325713037632, 6.56488065937706241961802387284, 7.326310871324706342831747327826, 8.43878039557817648077519528952, 9.06250413170943426753858943624, 9.55974071305251856465584961542, 10.470260932528618985455152027259, 11.27186712223807706129904351518, 11.55181664248422130238009313264, 12.283536504295707411335741800626, 13.19806529375946155104428956139, 14.220027924831867175392268348467, 15.410614046041383268996393132058, 15.50096675565435717533175998845, 16.21946623163932171154858271221, 16.939308614519440452298356775008, 17.66369689433716325908293951263, 18.23532089767703989015431826292, 18.64301453967916707982517763958

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