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Properties

Label 1-185-185.164-r0-0-0
Degree $1$
Conductor $185$
Sign $0.727 + 0.685i$
Analytic cond. $0.859136$
Root an. cond. $0.859136$
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  + (−0.766 − 0.642i)2-s + (−0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + 6-s + (0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.766 − 0.642i)12-s + (−0.173 − 0.984i)13-s + (−0.5 − 0.866i)14-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + (−0.766 + 0.642i)18-s + (0.766 − 0.642i)19-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + 6-s + (0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.766 − 0.642i)12-s + (−0.173 − 0.984i)13-s + (−0.5 − 0.866i)14-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + (−0.766 + 0.642i)18-s + (0.766 − 0.642i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.727 + 0.685i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.727 + 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5798551549 + 0.2302169752i\)
\(L(\frac12)\) \(\approx\) \(0.5798551549 + 0.2302169752i\)
\(L(1)\) \(\approx\) \(0.6364095376 + 0.06128814698i\)
\(L(1)\) \(\approx\) \(0.6364095376 + 0.06128814698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.173 + 0.984i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−26.95623922734004978996218943989, −26.43917910543776286778026870636, −24.7017120898331299728245449472, −24.551320026484986215565034448418, −23.53161234427071868768266637632, −22.74455331642152878458703515884, −21.29841295029998658211000094526, −20.18041865024001699211577901825, −18.68148754782276365390020121519, −18.57155727741345636633859105697, −17.28328515491394705125490370702, −16.666713977422841408203168661695, −15.71459405325739534997252776011, −14.21208387712723100455459976180, −13.61346103853193311544304728757, −11.80907676100559308310901533706, −11.16155255498722486153272092546, −10.10219834256792087159145745696, −8.63167937303018032000996195339, −7.67347912313547989892876827996, −6.8123356509411799240025818679, −5.62528839166010906877117876677, −4.69870637661737860752723319827, −2.15439040130736293837020209801, −0.80133074596235834147291759110, 1.35965451977178279081855615049, 2.96564424685060019752202611192, 4.441164332365063720463153537406, 5.46348991467277589048450546778, 7.15150996849266918485196472631, 8.25790175078826936203464705366, 9.47091376733923409472050659117, 10.40576615123060180846070024064, 11.20740421192187211069235741595, 12.122404930677683811727422233330, 13.08868745220287072037834509810, 15.013654942621498117545385751926, 15.65032432418142156133080296220, 17.012584464894496365895644720450, 17.72750979970926916805459886872, 18.2532359220506013486961101981, 19.796444205743371786849283356709, 20.65538276204509663383997133858, 21.46683728540492932632267992255, 22.24900054203709311579906188813, 23.344715456739176520717945552818, 24.56262765935759265612526105836, 25.745481412635585814782568090662, 26.679926870595547620757621055233, 27.57584332170686771827241737732

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