Properties

Label 1-185-185.127-r1-0-0
Degree $1$
Conductor $185$
Sign $0.966 - 0.258i$
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  + (−0.642 + 0.766i)2-s + (−0.642 − 0.766i)3-s + (−0.173 − 0.984i)4-s + 6-s + (0.342 − 0.939i)7-s + (0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.642 + 0.766i)12-s + (0.984 − 0.173i)13-s + (0.5 + 0.866i)14-s + (−0.939 + 0.342i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)18-s + (−0.766 + 0.642i)19-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.642 − 0.766i)3-s + (−0.173 − 0.984i)4-s + 6-s + (0.342 − 0.939i)7-s + (0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.642 + 0.766i)12-s + (0.984 − 0.173i)13-s + (0.5 + 0.866i)14-s + (−0.939 + 0.342i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)18-s + (−0.766 + 0.642i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.966 - 0.258i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.966 - 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9938519812 - 0.1306128015i\)
\(L(\frac12)\) \(\approx\) \(0.9938519812 - 0.1306128015i\)
\(L(1)\) \(\approx\) \(0.7033040804 + 0.005700290582i\)
\(L(1)\) \(\approx\) \(0.7033040804 + 0.005700290582i\)

Euler product

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

−27.345886673610022238671164817102, −26.26325517067215954106983572922, −25.53503541907878011827907017866, −24.05712583455639235498325855520, −22.93554894466139926697903418217, −21.819311953343195590777781257, −21.29887547916249169165374122944, −20.57199730999375215619175235083, −19.09827564153929219256713227082, −18.33356893290350562784350475866, −17.47892272993911941163867706885, −16.286527188650887867438631024845, −15.730691544270215141372536165384, −14.21509257560237693336207701368, −12.75798011461011393887069664356, −11.77748001645737063471571033238, −11.01254594401028992112443243501, −10.136115868671544735132745452826, −8.91601967400464353988621798908, −8.23991523062702801124781942264, −6.36033175702742859430346851094, −5.16811010338834436847843625021, −3.83751256754366245288345275477, −2.63386270490628954054125821699, −0.85085971077589220691615886615, 0.72498089801185228273341194381, 1.843765672351886560099416568638, 4.30881434643379244637645962001, 5.59870717406762052642444308724, 6.54399833165449891599570859224, 7.662096172686239094467269836585, 8.20317229245109758665481444975, 10.0564566949450459766491092764, 10.65773891420569162240747655129, 11.96783708385252835003641854848, 13.31978579774038752657462266685, 14.09568218032136888942923150400, 15.392980276780249222357068374966, 16.50675123241260710448641534473, 17.28094146518205075013578213664, 18.0345586496004873498543175555, 18.881587437968247493515066633731, 19.919005003182374204786518055640, 21.019601254526707872370021458678, 22.84781501125016261116686882911, 23.31402215759144993060659090818, 23.94131191489545047224523080105, 25.18883516573071852921091483124, 25.7567913865039971862039091315, 26.94455382171495548914875168716

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