Properties

Label 1-1045-1045.303-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.506 + 0.862i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (0.587 + 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.809 + 0.587i)6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + i·12-s + (−0.587 − 0.809i)13-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.587 − 0.809i)17-s + (0.951 + 0.309i)18-s + 21-s + i·23-s + (−0.809 + 0.587i)24-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.809 + 0.587i)6-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)8-s + (0.809 − 0.587i)9-s + i·12-s + (−0.587 − 0.809i)13-s + (0.309 + 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.587 − 0.809i)17-s + (0.951 + 0.309i)18-s + 21-s + i·23-s + (−0.809 + 0.587i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.506 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.376733142 + 2.503648523i\)
\(L(\frac12)\) \(\approx\) \(4.376733142 + 2.503648523i\)
\(L(1)\) \(\approx\) \(2.042054826 + 0.8299017664i\)
\(L(1)\) \(\approx\) \(2.042054826 + 0.8299017664i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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

−21.17933702973053878204279478207, −20.617431768204332275022552542023, −19.76271248027982659758232814466, −19.1642718439285388797724493482, −18.46371694705401481868633637564, −17.41986667409202086289368347575, −16.419377411294604734780089010932, −15.32404969425655521665710136606, −14.588858406437924866271872875967, −14.24742820138533057674855141333, −13.42316016434640201720091932479, −12.52780153567362552019794100386, −11.71564941192017762541434165167, −10.759231234694182051742899308976, −10.132383757395452933280326068548, −9.27982448224654665526156581979, −8.423379108206426509435246707693, −7.57357324016412404488402932701, −6.408085342231287786755390349405, −5.17463608734375261361383130068, −4.36350016011598953215030705709, −3.850801986157202428942151570032, −2.59876113399467500615616443171, −1.971962835654899849586294235717, −0.92759816031065061005226800215, 0.94784535982778488451864340693, 2.34511623533315346535960371311, 3.07294743608571335310571514017, 4.10618948792940570297097368942, 5.052419678978352386373309717384, 5.78859010610353856128526329862, 7.09218541753595719632244498364, 7.63551535687160850717209887401, 8.27964649741152410426125689678, 9.12405155261150091840221221407, 9.99774161900695079099042397022, 11.45395556280280896801421393002, 12.1763153649983734494348662049, 13.00225949486692284062350646813, 13.779319166788489916341414992066, 14.4472344928716974522409839513, 15.08670681370036868074662742139, 15.63170254777466402134397376789, 16.71618770833232688312824935542, 17.646789764070693679901365866539, 18.20253245487071141329981066538, 19.02824971551480467635664489304, 20.25013921630316969638993309222, 20.6507709782348764096133590694, 21.5842561392860443269678973014

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