↑

Properties

Label 1-2475-2475.1003-r0-0-0
Degree $1$
Conductor $2475$
Sign $0.995 + 0.0895i$
Analytic cond. $11.4938$
Root an. cond. $11.4938$
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  + (0.207 − 0.978i)2-s + (−0.913 − 0.406i)4-s + (0.207 − 0.978i)7-s + (−0.587 + 0.809i)8-s + (−0.866 + 0.5i)13-s + (−0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (−0.951 + 0.309i)17-s + (−0.809 − 0.587i)19-s + (−0.994 − 0.104i)23-s + (0.309 + 0.951i)26-s + (−0.587 + 0.809i)28-s + (−0.104 − 0.994i)29-s + (0.913 + 0.406i)31-s + (0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (−0.913 − 0.406i)4-s + (0.207 − 0.978i)7-s + (−0.587 + 0.809i)8-s + (−0.866 + 0.5i)13-s + (−0.913 − 0.406i)14-s + (0.669 + 0.743i)16-s + (−0.951 + 0.309i)17-s + (−0.809 − 0.587i)19-s + (−0.994 − 0.104i)23-s + (0.309 + 0.951i)26-s + (−0.587 + 0.809i)28-s + (−0.104 − 0.994i)29-s + (0.913 + 0.406i)31-s + (0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.995 + 0.0895i$
Analytic conductor: \(11.4938\)
Root analytic conductor: \(11.4938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (0:\ ),\ 0.995 + 0.0895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7267843332 + 0.03262374836i\)
\(L(\frac12)\) \(\approx\) \(0.7267843332 + 0.03262374836i\)
\(L(1)\) \(\approx\) \(0.7416479756 - 0.4241316273i\)
\(L(1)\) \(\approx\) \(0.7416479756 - 0.4241316273i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.207 - 0.978i)T \)
7 \( 1 + (0.207 - 0.978i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.207 + 0.978i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.207 + 0.978i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.743 - 0.669i)T \)
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

−19.22522028747356729088591175818, −18.64158805295058686832698250696, −17.75084378021995584994358982282, −17.488386153115727539802803243518, −16.4846173558979489782059504920, −15.817968504652298261357777111605, −15.171042594227940992900278831565, −14.66930216796913030851612108263, −13.880699173635203153614358631376, −13.09642590900478294923245313461, −12.25528944917730308927366153046, −11.92732818727885865895855786858, −10.60943217071881036717526771867, −9.884092434299943535354227741653, −8.84696553021042298696550096745, −8.59631002000868796782513231914, −7.58617382544732112895091466539, −6.95331953629808195793085080554, −5.96117010215578690836155603972, −5.49734312105660736179352229611, −4.63164304218758593375045120234, −3.88372011881090552171577766548, −2.75252685418693620753052892715, −1.960372630343319446756098048111, −0.257758878137033247698630936048, 0.89446904650112142151316810548, 1.9863534492138911066987048870, 2.58447794138607291734295760435, 3.75583842967163893159347262804, 4.42263693253132088014187422337, 4.86168353072903416856488963282, 6.134565608579546245075283958350, 6.83902936037981131681365392695, 7.90801664652696427278160474558, 8.60139709471359077957798580531, 9.53311491653983873059644562988, 10.16905439559322431179811295561, 10.79648136399521945228785099110, 11.543754647500221407158950216058, 12.174665856585747568316630154479, 13.07840190948200759467258173960, 13.62507268333066977172861015876, 14.263799802921238916711945293255, 14.99954402509582121310733691480, 15.83514761450941401760045477627, 17.0599010028654071744199978620, 17.33409986606442860584102848085, 18.07211902703925840967811056749, 19.16049254013096500487808968070, 19.459993353384592718328754651047

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