Properties

Label 1-185-185.179-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

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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: $\chi_{185} (179, \cdot )$
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 \( 1 + iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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