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Properties

Label 1-185-185.83-r1-0-0
Degree $1$
Conductor $185$
Sign $0.863 + 0.504i$
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  + (0.342 + 0.939i)2-s + (0.342 − 0.939i)3-s + (−0.766 + 0.642i)4-s + 6-s + (0.984 − 0.173i)7-s + (−0.866 − 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.342 + 0.939i)12-s + (0.642 + 0.766i)13-s + (0.5 + 0.866i)14-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + (0.342 − 0.939i)18-s + (0.939 + 0.342i)19-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (0.342 − 0.939i)3-s + (−0.766 + 0.642i)4-s + 6-s + (0.984 − 0.173i)7-s + (−0.866 − 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.342 + 0.939i)12-s + (0.642 + 0.766i)13-s + (0.5 + 0.866i)14-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + (0.342 − 0.939i)18-s + (0.939 + 0.342i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.863 + 0.504i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.863 + 0.504i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.460326105 + 0.6668105238i\)
\(L(\frac12)\) \(\approx\) \(2.460326105 + 0.6668105238i\)
\(L(1)\) \(\approx\) \(1.470846577 + 0.3394072866i\)
\(L(1)\) \(\approx\) \(1.470846577 + 0.3394072866i\)

Euler product

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

−27.13848004903082126849325166526, −26.31009089807919864768086890939, −24.962422571710968230059433361094, −23.797548498583415137919906551218, −22.857992083807308266965648370353, −21.70264289520853781457971037664, −21.19992176263328212457992015370, −20.443481045548341718574469885987, −19.44726273119207582070633656551, −18.35863652231004651909627426984, −17.336860564743060238996729444476, −15.88440556952594917564356200704, −14.95332430676551457130149565062, −14.06587018474188240005534629522, −13.161103229941794493408093039699, −11.66896314232889303884263983046, −10.89856965832477291516552823854, −10.12212053763246949703620479151, −8.80219562692880015885221595157, −8.099402962230463474029440397856, −5.64787459993405800463780930957, −5.01033684996259965624694393765, −3.64552743687174386099100035480, −2.78192838926539079503707284911, −1.09584137482545198114888876525, 1.076308038105180012248533687925, 2.74109338822047665977346584123, 4.340656222626907202315358620466, 5.48205182416641380320059165447, 6.78388323492455078498524075988, 7.61663996032001520525051275779, 8.39709260298759007158863148532, 9.61178758620711182322614311216, 11.50539832435981241067838383896, 12.385647697531028129876220435826, 13.56692050789751076010975888221, 14.190636440166900385751758314957, 15.09036860790921258303299476502, 16.30914456872960998033362134067, 17.500089904975901523343523571841, 18.13101970244919399084672046839, 18.979536724417883515180278677226, 20.61560485687124428758165776036, 21.1253507052905206991039401798, 22.941681524779236324868022951625, 23.226401525642346662607454182546, 24.42942637004873998905235752335, 24.86416100527513219827373756521, 25.956108302584983634256825118507, 26.67256222743183492203288668729

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