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Properties

Label 1-185-185.22-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: $\chi_{185} (22, \cdot )$
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

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

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