↑

Properties

Label 1-185-185.154-r1-0-0
Degree $1$
Conductor $185$
Sign $0.646 - 0.763i$
Analytic cond. $19.8810$
Root an. cond. $19.8810$
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  + i·2-s + 3-s − 4-s + i·6-s − 7-s − i·8-s + 9-s − 11-s − 12-s − i·13-s − i·14-s + 16-s − i·17-s + i·18-s + i·19-s + ⋯
L(s)  = 1  + i·2-s + 3-s − 4-s + i·6-s − 7-s − i·8-s + 9-s − 11-s − 12-s − i·13-s − i·14-s + 16-s − i·17-s + i·18-s + i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.646 - 0.763i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.646 - 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.071467423 - 0.4965764825i\)
\(L(\frac12)\) \(\approx\) \(1.071467423 - 0.4965764825i\)
\(L(1)\) \(\approx\) \(1.003544809 + 0.2447081406i\)
\(L(1)\) \(\approx\) \(1.003544809 + 0.2447081406i\)

Euler product

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

−27.012063060241994055196046506466, −26.09681371136833183657944309956, −25.74072773077280539837078818539, −24.02360507520730722825800839178, −23.31227923758538369810355429887, −21.77899058981449764855844475262, −21.45157134021878552176626568685, −20.206196989212916111277243687010, −19.4738268572190244347063014781, −18.85379485516285009730126644046, −17.77780994898622979973101910636, −16.28324577366979291691632839903, −15.203117721865183739742770555875, −13.99202338221606697797341953163, −13.17605538119554045691501260519, −12.5035491253095662799268504493, −11.007759203418315072664564662832, −9.94378552917685705191443504244, −9.17541849340555390362445666405, −8.1798146344187310167283826596, −6.78163065324966135577365174960, −4.98403212920268184694196493009, −3.67977132826092834553741285142, −2.83910338222353640482801701319, −1.60794880424206541462915900830, 0.36146586693446513200684428813, 2.68738777027730798978542437459, 3.78391439491946950807281220064, 5.19774479024915329191412666986, 6.46939405675856862271074759901, 7.63407560049569637485535907470, 8.3428412629558852174656691832, 9.62057688902412224809883724818, 10.233368103737733955933389114909, 12.54368136488237497949773424847, 13.23296689678351706288976944108, 14.10594301430308687103712776056, 15.297674756137726687068613241434, 15.81827790178942657235807510394, 16.853103909921250913268708639722, 18.39451727537422162024688417607, 18.76924407731746335231150112115, 20.09985747718392549194869031093, 21.01307241543923697594924578588, 22.39184412849385960028981082353, 23.03479818472864281949961248579, 24.29397122703700359660344233662, 25.12163611593391903592564683072, 25.753265210910249817983244081171, 26.60732907933230834894552724215

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