↑

Properties

Label 1-575-575.234-r0-0-0
Degree $1$
Conductor $575$
Sign $0.906 - 0.422i$
Analytic cond. $2.67028$
Root an. cond. $2.67028$
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  + (0.921 + 0.389i)2-s + (0.362 − 0.931i)3-s + (0.696 + 0.717i)4-s + (0.696 − 0.717i)6-s + (0.959 − 0.281i)7-s + (0.362 + 0.931i)8-s + (−0.736 − 0.676i)9-s + (0.516 − 0.856i)11-s + (0.921 − 0.389i)12-s + (0.564 − 0.825i)13-s + (0.993 + 0.113i)14-s + (−0.0285 + 0.999i)16-s + (−0.696 + 0.717i)17-s + (−0.415 − 0.909i)18-s + (−0.466 + 0.884i)19-s + ⋯
L(s)  = 1  + (0.921 + 0.389i)2-s + (0.362 − 0.931i)3-s + (0.696 + 0.717i)4-s + (0.696 − 0.717i)6-s + (0.959 − 0.281i)7-s + (0.362 + 0.931i)8-s + (−0.736 − 0.676i)9-s + (0.516 − 0.856i)11-s + (0.921 − 0.389i)12-s + (0.564 − 0.825i)13-s + (0.993 + 0.113i)14-s + (−0.0285 + 0.999i)16-s + (−0.696 + 0.717i)17-s + (−0.415 − 0.909i)18-s + (−0.466 + 0.884i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $0.906 - 0.422i$
Analytic conductor: \(2.67028\)
Root analytic conductor: \(2.67028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 575,\ (0:\ ),\ 0.906 - 0.422i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.975715551 - 0.6596848079i\)
\(L(\frac12)\) \(\approx\) \(2.975715551 - 0.6596848079i\)
\(L(1)\) \(\approx\) \(2.149025366 - 0.2192109264i\)
\(L(1)\) \(\approx\) \(2.149025366 - 0.2192109264i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.921 + 0.389i)T \)
3 \( 1 + (0.362 - 0.931i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
11 \( 1 + (0.516 - 0.856i)T \)
13 \( 1 + (0.564 - 0.825i)T \)
17 \( 1 + (-0.696 + 0.717i)T \)
19 \( 1 + (-0.466 + 0.884i)T \)
29 \( 1 + (-0.466 - 0.884i)T \)
31 \( 1 + (-0.362 - 0.931i)T \)
37 \( 1 + (0.736 + 0.676i)T \)
41 \( 1 + (0.993 - 0.113i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.610 + 0.791i)T \)
59 \( 1 + (-0.564 + 0.825i)T \)
61 \( 1 + (-0.254 + 0.967i)T \)
67 \( 1 + (-0.974 + 0.226i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (-0.897 + 0.441i)T \)
79 \( 1 + (0.941 - 0.336i)T \)
83 \( 1 + (0.870 - 0.491i)T \)
89 \( 1 + (0.774 + 0.633i)T \)
97 \( 1 + (0.870 + 0.491i)T \)
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

−23.18764588757512998045385400086, −22.150310166744499086048235883786, −21.688478614935951733388198916732, −20.87646100386989095996540689796, −20.23848941732239993547378118206, −19.56956962873216219422303014603, −18.385458803019521939337262904759, −17.34226723534984835273359068324, −16.19414814513154530232944016240, −15.52776630458891876903887657684, −14.63678157410903849274062048495, −14.23598683989241418569895545196, −13.23746787756108333385540336421, −12.0866280377650283876811272287, −11.18982605094273098461897446058, −10.78387807836985106231429894025, −9.422920415243219397101799998425, −8.89975204592600850090512184970, −7.44521806463435863017464620618, −6.40707376868744739564643662321, −5.0928671092899727304716038535, −4.59899077412619569194419486753, −3.76459398660274470836011225087, −2.51110532040182070470486732324, −1.68579039366407192127908422506, 1.28965242796024028221103322011, 2.32020775865569731949416657192, 3.508974170916114695902122637499, 4.34053842523346237101148438519, 5.8751752653661555243515838330, 6.16257302982029789734223926998, 7.52863066831880600832348857763, 8.07921390533272840665506754337, 8.83167984335215130355028822042, 10.70491040331283980528043293678, 11.41947136022687981793559969245, 12.24818594271905777395953496719, 13.315891535761155980933994895737, 13.6400753532689770956416689293, 14.77205812705389614038332526211, 15.085276708248469675870277686941, 16.54087708019140184870280700291, 17.24262440320998658990612835382, 18.04881177722686135465287009878, 19.06508113823960175066173931976, 20.08210232270120148698523479635, 20.67389202791429644366916238088, 21.53385055546274111259693709558, 22.53205735453009205232065588383, 23.370009473329310695655955563898

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