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Properties

Label 1-185-185.128-r0-0-0
Degree $1$
Conductor $185$
Sign $0.999 + 0.00624i$
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  + (0.939 − 0.342i)2-s + (−0.342 + 0.939i)3-s + (0.766 − 0.642i)4-s + i·6-s + (0.984 − 0.173i)7-s + (0.5 − 0.866i)8-s + (−0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (0.342 + 0.939i)12-s + (−0.766 + 0.642i)13-s + (0.866 − 0.5i)14-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (−0.939 − 0.342i)18-s + (−0.342 + 0.939i)19-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (−0.342 + 0.939i)3-s + (0.766 − 0.642i)4-s + i·6-s + (0.984 − 0.173i)7-s + (0.5 − 0.866i)8-s + (−0.766 − 0.642i)9-s + (0.5 − 0.866i)11-s + (0.342 + 0.939i)12-s + (−0.766 + 0.642i)13-s + (0.866 − 0.5i)14-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (−0.939 − 0.342i)18-s + (−0.342 + 0.939i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.999 + 0.00624i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.999 + 0.00624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.924940265 + 0.006012641163i\)
\(L(\frac12)\) \(\approx\) \(1.924940265 + 0.006012641163i\)
\(L(1)\) \(\approx\) \(1.692466203 + 0.009403137857i\)
\(L(1)\) \(\approx\) \(1.692466203 + 0.009403137857i\)

Euler product

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

−27.293760887072240693624515754247, −25.79588023073830272856879642300, −24.911505485619032831144501738653, −24.41307035293562318934701099648, −23.40433080980754122979260513919, −22.664756858131987193073151211814, −21.73783794777798558664161553259, −20.51363613791902271803031200799, −19.73786882215556079922333865664, −18.266329905415554459663959080581, −17.4121675732776535166407180760, −16.64953170903082168245897180759, −15.055367684692007167418364329165, −14.49598532793796757966652872485, −13.362346083818104661854796485504, −12.340600423282270314465632979345, −11.764327124639444664959380095232, −10.616078097679487063312453421849, −8.629287978835260773387302339811, −7.464340991828687072836325143115, −6.85538229578964000177701991679, −5.38592283886597652769169656939, −4.73434300387195266483416682125, −2.85517749050557000933821514600, −1.688161876302060898965429078850, 1.62780739781386330706988651754, 3.37301404624607489563138589003, 4.2787268780877096892806615947, 5.31188619210369535870012509572, 6.23746806118938136688627802445, 7.861266622488960517694749328797, 9.410060578999306616566394812690, 10.475187754300322257127560971131, 11.42928115356021886956176387762, 12.007522566790003986436404951357, 13.58373573653588825117572400268, 14.6452056498133072221874142172, 15.03496813912523779809376195923, 16.57017565208584818374358853914, 17.01787819740321809042134919419, 18.754425900630745559425174642698, 19.80889537391046835435702647690, 20.97460659599785040277737379739, 21.40132938849807196405010351559, 22.242508798870946070861436866419, 23.32749086347605231755235382305, 24.0621565204708205384395650554, 25.07217955511810766406389003709, 26.44280738764725021130340325014, 27.43211069188340249237852378247

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