↑

Properties

Label 1-55e2-3025.1581-r1-0-0
Degree $1$
Conductor $3025$
Sign $0.989 + 0.145i$
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  + (−0.897 + 0.441i)2-s + (0.309 + 0.951i)3-s + (0.610 − 0.791i)4-s + (−0.696 − 0.717i)6-s + (0.362 + 0.931i)7-s + (−0.198 + 0.980i)8-s + (−0.809 + 0.587i)9-s + (0.941 + 0.336i)12-s + (−0.610 + 0.791i)13-s + (−0.736 − 0.676i)14-s + (−0.254 − 0.967i)16-s + (−0.974 − 0.226i)17-s + (0.466 − 0.884i)18-s + (−0.974 + 0.226i)19-s + (−0.774 + 0.633i)21-s + ⋯
L(s)  = 1  + (−0.897 + 0.441i)2-s + (0.309 + 0.951i)3-s + (0.610 − 0.791i)4-s + (−0.696 − 0.717i)6-s + (0.362 + 0.931i)7-s + (−0.198 + 0.980i)8-s + (−0.809 + 0.587i)9-s + (0.941 + 0.336i)12-s + (−0.610 + 0.791i)13-s + (−0.736 − 0.676i)14-s + (−0.254 − 0.967i)16-s + (−0.974 − 0.226i)17-s + (0.466 − 0.884i)18-s + (−0.974 + 0.226i)19-s + (−0.774 + 0.633i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.989 + 0.145i$
Analytic conductor: \(325.081\)
Root analytic conductor: \(325.081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1581, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (1:\ ),\ 0.989 + 0.145i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01957736068 + 0.001430857814i\)
\(L(\frac12)\) \(\approx\) \(0.01957736068 + 0.001430857814i\)
\(L(1)\) \(\approx\) \(0.4256483027 + 0.4043991389i\)
\(L(1)\) \(\approx\) \(0.4256483027 + 0.4043991389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.897 + 0.441i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.362 + 0.931i)T \)
13 \( 1 + (-0.610 + 0.791i)T \)
17 \( 1 + (-0.974 - 0.226i)T \)
19 \( 1 + (-0.974 + 0.226i)T \)
23 \( 1 + (-0.362 + 0.931i)T \)
29 \( 1 + (-0.0855 + 0.996i)T \)
31 \( 1 + (-0.959 - 0.281i)T \)
37 \( 1 + (-0.0285 - 0.999i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (-0.985 + 0.170i)T \)
53 \( 1 + (-0.362 - 0.931i)T \)
59 \( 1 + (-0.985 + 0.170i)T \)
61 \( 1 + (-0.696 + 0.717i)T \)
67 \( 1 + (0.897 - 0.441i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (0.870 - 0.491i)T \)
83 \( 1 + (0.998 - 0.0570i)T \)
89 \( 1 + (0.974 + 0.226i)T \)
97 \( 1 + (0.198 - 0.980i)T \)
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

−18.13841401881336879507786600276, −17.50497634733577777401388923474, −17.165894939705697623370215266717, −16.392819646434001163533997929277, −15.2750995465897527154706976197, −14.730271980872549199813724783013, −13.71695194428771055936669058228, −13.07728049784187440020189144356, −12.5303584694106395996546637434, −11.75534151941776667670671839495, −10.89862877641547698552218773557, −10.487123840145467287495208061581, −9.51154313223495963760633656845, −8.71995664096780350824102576202, −7.99069394681441784643672639451, −7.6216845465676794884905880836, −6.67381243186560557688214770036, −6.27452561961385495781000814967, −4.80526689205125467246389513951, −3.90990693109496971955680946901, −2.97899221162007954423460833608, −2.18765852261518931019917787030, −1.53625224736677561163355292310, −0.4603349595077291127902344495, −0.00699658371447848058309101408, 1.81237405584821758840736805161, 2.16491420936867404069005976760, 3.2216566749124978878568446184, 4.376771747798722059365232214250, 5.07670533261989600736227309130, 5.78248713803060621960092100827, 6.623368327804828918902663288230, 7.548518683109828892207745429104, 8.33242539870563404789704807912, 9.05068552373141432733048893523, 9.31966726942521877583838477415, 10.17698413792088147935060126967, 11.05406299347855286106992910137, 11.424941896479993851352048775838, 12.33508869921591836861249237386, 13.491783879030071602895238238922, 14.548250445740562193451074444056, 14.72442797838736561123057782786, 15.49337713560692197523727027991, 16.13677354654462204869540086904, 16.677712058839979433833219239946, 17.5010783264407762140214069016, 18.07504949713714900808485405700, 18.91670025730098092758644961401, 19.56509870551032284490035640075

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