↑

Properties

Label 1-1832-1832.589-r1-0-0
Degree $1$
Conductor $1832$
Sign $0.703 + 0.710i$
Analytic cond. $196.875$
Root an. cond. $196.875$
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  + (0.754 − 0.656i)3-s + (−0.998 − 0.0550i)5-s + (0.218 − 0.975i)7-s + (0.137 − 0.990i)9-s + (−0.986 + 0.164i)11-s + (−0.837 − 0.546i)13-s + (−0.789 + 0.614i)15-s + (0.546 − 0.837i)17-s + (−0.451 + 0.892i)19-s + (−0.475 − 0.879i)21-s + (0.569 − 0.821i)23-s + (0.993 + 0.110i)25-s + (−0.546 − 0.837i)27-s + (−0.954 + 0.298i)29-s + (−0.771 + 0.635i)31-s + ⋯
L(s)  = 1  + (0.754 − 0.656i)3-s + (−0.998 − 0.0550i)5-s + (0.218 − 0.975i)7-s + (0.137 − 0.990i)9-s + (−0.986 + 0.164i)11-s + (−0.837 − 0.546i)13-s + (−0.789 + 0.614i)15-s + (0.546 − 0.837i)17-s + (−0.451 + 0.892i)19-s + (−0.475 − 0.879i)21-s + (0.569 − 0.821i)23-s + (0.993 + 0.110i)25-s + (−0.546 − 0.837i)27-s + (−0.954 + 0.298i)29-s + (−0.771 + 0.635i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ 0.703 + 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3455052141 + 0.1441355184i\)
\(L(\frac12)\) \(\approx\) \(0.3455052141 + 0.1441355184i\)
\(L(1)\) \(\approx\) \(0.8175276516 - 0.3756060543i\)
\(L(1)\) \(\approx\) \(0.8175276516 - 0.3756060543i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (0.754 - 0.656i)T \)
5 \( 1 + (-0.998 - 0.0550i)T \)
7 \( 1 + (0.218 - 0.975i)T \)
11 \( 1 + (-0.986 + 0.164i)T \)
13 \( 1 + (-0.837 - 0.546i)T \)
17 \( 1 + (0.546 - 0.837i)T \)
19 \( 1 + (-0.451 + 0.892i)T \)
23 \( 1 + (0.569 - 0.821i)T \)
29 \( 1 + (-0.954 + 0.298i)T \)
31 \( 1 + (-0.771 + 0.635i)T \)
37 \( 1 + (-0.0275 + 0.999i)T \)
41 \( 1 + (-0.376 + 0.926i)T \)
43 \( 1 + (0.879 - 0.475i)T \)
47 \( 1 + (-0.990 - 0.137i)T \)
53 \( 1 + (-0.945 - 0.324i)T \)
59 \( 1 + (0.999 - 0.0275i)T \)
61 \( 1 + (-0.789 - 0.614i)T \)
67 \( 1 + (-0.376 - 0.926i)T \)
71 \( 1 + (-0.350 - 0.936i)T \)
73 \( 1 + (-0.110 + 0.993i)T \)
79 \( 1 + (0.954 + 0.298i)T \)
83 \( 1 + (-0.851 + 0.523i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.592 + 0.805i)T \)
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

−19.6996653042851417100056070649, −19.18566748799102652893530623179, −18.80752437270034334916316924269, −17.706185078771276722518175502917, −16.71161663202969994430443855420, −16.01358961522790840383602641116, −15.26398992678703599952125280511, −14.98252652809251106995982167609, −14.24294495888702505199227329619, −13.04947546263307778117747186792, −12.58771995866731484951195208572, −11.440500511791339040728483810158, −11.022535973789931714373272333648, −10.03232100717630388227558577495, −9.12719456103298866241492879302, −8.64285071897467765866838653762, −7.68287740253151769443223559474, −7.34344133685111645977568833302, −5.81270386648522318338635417466, −5.05647972458030053359193843199, −4.30051242650483893166048678427, −3.41398194166300639019813806702, −2.62935040600698188627962088936, −1.862209539451544864641979077215, −0.07979821286111338880351223567, 0.68971700709939301351842413623, 1.760092965531913463832253304325, 2.93880151590380005238490084802, 3.446542879012305519551841438195, 4.504709074320052295258503564624, 5.231622744817859929982480350042, 6.65302654062876108639472349998, 7.34967477313992315018926195721, 7.84182745782423519733329058562, 8.35936268447677839343885368824, 9.50164826012472987562162979320, 10.316410333941979239561831419883, 11.07488990915206135158898128488, 12.07382241324957270128728383214, 12.717444112828077531323243752204, 13.24095511140126385919840051326, 14.36155359015656824612079153967, 14.68962071273674275098931938700, 15.51417590482638889192550551295, 16.43167314687155435488514106948, 17.05745676776875809654155200096, 18.17231177142518850280972243081, 18.607638377351930005443757837391, 19.39521259921669633657579446919, 20.10647217629341361475939649243

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