↑

Properties

Label 1-1800-1800.347-r1-0-0
Degree $1$
Conductor $1800$
Sign $-0.916 + 0.400i$
Analytic cond. $193.436$
Root an. cond. $193.436$
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  + (0.866 + 0.5i)7-s + (−0.913 + 0.406i)11-s + (−0.406 + 0.913i)13-s + (−0.951 − 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.994 + 0.104i)23-s + (0.978 + 0.207i)29-s + (0.978 − 0.207i)31-s + (0.587 + 0.809i)37-s + (−0.913 − 0.406i)41-s + (0.866 + 0.5i)43-s + (0.207 − 0.978i)47-s + (0.5 + 0.866i)49-s + (0.951 − 0.309i)53-s + (0.913 + 0.406i)59-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (−0.913 + 0.406i)11-s + (−0.406 + 0.913i)13-s + (−0.951 − 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.994 + 0.104i)23-s + (0.978 + 0.207i)29-s + (0.978 − 0.207i)31-s + (0.587 + 0.809i)37-s + (−0.913 − 0.406i)41-s + (0.866 + 0.5i)43-s + (0.207 − 0.978i)47-s + (0.5 + 0.866i)49-s + (0.951 − 0.309i)53-s + (0.913 + 0.406i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.916 + 0.400i$
Analytic conductor: \(193.436\)
Root analytic conductor: \(193.436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ -0.916 + 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2680508032 + 1.283086824i\)
\(L(\frac12)\) \(\approx\) \(0.2680508032 + 1.283086824i\)
\(L(1)\) \(\approx\) \(0.9874937533 + 0.2705301182i\)
\(L(1)\) \(\approx\) \(0.9874937533 + 0.2705301182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.994 + 0.104i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.913 - 0.406i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.207 - 0.978i)T \)
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

−19.73691295269342892981256396188, −19.03613080265764749076059250466, −17.97463160225124048424047939139, −17.596115869731354186953090871463, −16.94051744578271007078626535168, −15.825224770114769769319146863383, −15.3178361365883364191513059287, −14.57760006345246200817340993835, −13.59472122835415236836448753526, −13.164657390291053881432322840457, −12.2710619382698482089190981332, −11.21633650365099853236631505842, −10.75896675225957031057601201272, −10.10247756647609440894921532538, −8.899546675451393148738644732848, −8.28653403510872881707268201274, −7.53209790498262226227636384311, −6.755776148188261367872773111794, −5.698062228304393311974519540848, −4.85562847115206517773984352005, −4.316223758960526334810530600957, −2.97082007624397638190430244826, −2.385440440749636231774032551279, −1.06376393676510245599160402146, −0.25367814957670586375968750836, 1.18955853643947675277386797014, 2.19140935990134184217383749960, 2.78329025572437077490530842127, 4.23258126969288689639953817048, 4.782887174776265458547690786725, 5.568715023993554883731455743526, 6.625432736905731094005452275877, 7.35602238421045823845908187903, 8.305024963054103772113525105351, 8.81749707692406092108084269996, 9.85273515938749509341782774070, 10.55377545968032416196678522632, 11.486666152578176985481656063937, 11.98569866870057613050957861073, 12.89285511162796880698318798341, 13.689879696724751074533916896, 14.46519121516187844131933285241, 15.20054496707738660490057398979, 15.74828011982858354314932604119, 16.76706173516850851079517618527, 17.397921701445443138737723800550, 18.2455512404763494490929383355, 18.69523105760995763862140736622, 19.572367398495224965524704739392, 20.448080517200606087274243687446

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