↑

Properties

Label 1-605-605.197-r0-0-0
Degree $1$
Conductor $605$
Sign $-0.0435 - 0.999i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (0.989 + 0.142i)2-s − i·3-s + (0.959 + 0.281i)4-s + (0.142 − 0.989i)6-s + (−0.540 − 0.841i)7-s + (0.909 + 0.415i)8-s − 9-s + (0.281 − 0.959i)12-s + (0.281 − 0.959i)13-s + (−0.415 − 0.909i)14-s + (0.841 + 0.540i)16-s + (0.755 + 0.654i)17-s + (−0.989 − 0.142i)18-s + (−0.654 − 0.755i)19-s + (−0.841 + 0.540i)21-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)2-s − i·3-s + (0.959 + 0.281i)4-s + (0.142 − 0.989i)6-s + (−0.540 − 0.841i)7-s + (0.909 + 0.415i)8-s − 9-s + (0.281 − 0.959i)12-s + (0.281 − 0.959i)13-s + (−0.415 − 0.909i)14-s + (0.841 + 0.540i)16-s + (0.755 + 0.654i)17-s + (−0.989 − 0.142i)18-s + (−0.654 − 0.755i)19-s + (−0.841 + 0.540i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0435 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0435 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0435 - 0.999i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ -0.0435 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.639529271 - 1.712596877i\)
\(L(\frac12)\) \(\approx\) \(1.639529271 - 1.712596877i\)
\(L(1)\) \(\approx\) \(1.634114067 - 0.7108588913i\)
\(L(1)\) \(\approx\) \(1.634114067 - 0.7108588913i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.989 + 0.142i)T \)
3 \( 1 - iT \)
7 \( 1 + (-0.540 - 0.841i)T \)
13 \( 1 + (0.281 - 0.959i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (0.540 - 0.841i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + (-0.281 - 0.959i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (0.989 - 0.142i)T \)
53 \( 1 + (-0.540 - 0.841i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (-0.989 - 0.142i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.540 + 0.841i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (0.540 + 0.841i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (0.909 + 0.415i)T \)
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

−23.19903770079321128277757134644, −22.201424046186879501097148448313, −21.80285210480297786274625683180, −20.954463781895051029034313348727, −20.40293793577377641757520810153, −19.24551414952024351765093617172, −18.65417243282576539146041117137, −17.012216955916418603925319987056, −16.326294983455094482486975064110, −15.707542957357711659144163230476, −14.82641181228194364204526412819, −14.24885883747053907101673105501, −13.19652201638950629388022083401, −12.19251609924328520925970372094, −11.51302703188497540176173052356, −10.650870865284104058210289918853, −9.62621377365952828127300247839, −8.94765107299123623976531274679, −7.54192943765072542636024654243, −6.29282202478482006808962680593, −5.603172899569560129833261312534, −4.734405790248416538434535725195, −3.679749935442255380754280266857, −3.00928430204388858256852201229, −1.80540021802366883629274607016, 0.85561908676491271075330798191, 2.20746492898517353768622290330, 3.20348739742475592233508281028, 4.11098405240074364210268323013, 5.49013110451567716381573887737, 6.1782482283121860668506352644, 7.147281112605943699171957925388, 7.699559385258373293063736798645, 8.828918204161007235588755852214, 10.48326832266927479183049757762, 10.99021901600290052704558963376, 12.2535512646166677444375125005, 12.883154456434096517429352886980, 13.37140017552054057044854513362, 14.334800190687299228895067349505, 15.047724521683087611075684726603, 16.192953840011025183864485039137, 17.0109278267347147129547108842, 17.671061615177274555327258047481, 19.00707097184041888903136605311, 19.61647582825098968666354003710, 20.41310291194846384687548578013, 21.16435029216596220759319921332, 22.54878451141948544311335868896, 22.78236563183671611835078969448

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