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Properties

Label 1-185-185.143-r0-0-0
Degree $1$
Conductor $185$
Sign $-0.133 - 0.991i$
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 + (0.642 − 0.766i)3-s + (0.173 − 0.984i)4-s − i·6-s + (0.342 + 0.939i)7-s + (−0.5 − 0.866i)8-s + (−0.173 − 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.642 − 0.766i)12-s + (0.173 − 0.984i)13-s + (0.866 + 0.5i)14-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (−0.766 − 0.642i)18-s + (−0.642 + 0.766i)19-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.642 − 0.766i)3-s + (0.173 − 0.984i)4-s − i·6-s + (0.342 + 0.939i)7-s + (−0.5 − 0.866i)8-s + (−0.173 − 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.642 − 0.766i)12-s + (0.173 − 0.984i)13-s + (0.866 + 0.5i)14-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (−0.766 − 0.642i)18-s + (−0.642 + 0.766i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.133 - 0.991i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ -0.133 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.369920735 - 1.566734535i\)
\(L(\frac12)\) \(\approx\) \(1.369920735 - 1.566734535i\)
\(L(1)\) \(\approx\) \(1.495190214 - 1.030542801i\)
\(L(1)\) \(\approx\) \(1.495190214 - 1.030542801i\)

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 \( 1 + (0.642 - 0.766i)T \)
7 \( 1 + (0.342 + 0.939i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.342 - 0.939i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (-0.342 - 0.939i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.342 - 0.939i)T \)
83 \( 1 + (-0.984 + 0.173i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
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

−27.04269819491873463705053670839, −26.32100894403911542329502817025, −25.79416236561850072372228715729, −24.34498645795198754259742199238, −23.93533730814338369398201703487, −22.58240923194822185983720774901, −21.73384252983103338737607811137, −20.99870908192133279706872546967, −20.081191296990690374546933447187, −18.97645157651028879935774659546, −17.130395575321985323385712980421, −16.719582041754946231875153758710, −15.61750895034141852004464383322, −14.60014351209783046201638674096, −13.94121732748936252687873816058, −13.1203386306663876465820942158, −11.49154957934227999543045734787, −10.63263958773905955739257729741, −9.03633655343981880306095893685, −8.234200694931209234218628619358, −7.02522722536625848513729645529, −5.79611811015211666203011907760, −4.169631124216759172862447850551, −3.999963447219478893377648399613, −2.30630424396782458150208733218, 1.51643396374648654881898178152, 2.49672260166137446394130309028, 3.6592642779672555949761273717, 5.17148184641009340630873054851, 6.26756224468457048393792756318, 7.53432418067640924833302831795, 8.88045419006179747999528129593, 9.86422532547662095165001114046, 11.37872366588714873394546738050, 12.302142589379798142659205962192, 12.91570603066697363931596565844, 14.14672304076761273856050134517, 14.861503245584391237460086250524, 15.71263579462816652445592498682, 17.73646839724341402398642141425, 18.410618768917442174119833556431, 19.43877416018606699114065101403, 20.29217216979610563032738715454, 21.0142153497828836421054441038, 22.2302521804391559998129576341, 23.07530799520999390438084005524, 24.07732079067951830153443109204, 25.13115817897627441770094416962, 25.37977154495520241343205697179, 27.27229185976684372757806631470

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