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Properties

Label 1-185-185.139-r0-0-0
Degree $1$
Conductor $185$
Sign $0.908 + 0.416i$
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 + (0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s − 6-s + (−0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.939 − 0.342i)12-s + (0.766 − 0.642i)13-s + (0.5 + 0.866i)14-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (−0.939 − 0.342i)18-s + (0.939 + 0.342i)19-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s − 6-s + (−0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.939 − 0.342i)12-s + (0.766 − 0.642i)13-s + (0.5 + 0.866i)14-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (−0.939 − 0.342i)18-s + (0.939 + 0.342i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.908 + 0.416i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.908 + 0.416i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.039979399 + 0.2271826279i\)
\(L(\frac12)\) \(\approx\) \(1.039979399 + 0.2271826279i\)
\(L(1)\) \(\approx\) \(0.9658718764 + 0.1662442784i\)
\(L(1)\) \(\approx\) \(0.9658718764 + 0.1662442784i\)

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

−27.07303320497429834842733733120, −26.108476297167925441143040957851, −25.498940712137914492178006982078, −24.63311751334599891903920341914, −23.70749705638871777051770194243, −21.85254630688313344856314387405, −21.21114132399546723839009237080, −20.26329735699565143886925218142, −19.24693987522914783721316620125, −18.53977484119399075071888673501, −17.98261900109636221286900565210, −16.17309162528786896287649732189, −15.79546801331167865732207480853, −14.347232596907585339508358469777, −13.23773834606081234743442862949, −12.14056793045747733119458315623, −11.19656139872043326815464558613, −9.70408953341110139248830453412, −8.992170890038566354521093087657, −8.13367933501709228578003116376, −7.09562784273645727815303699, −5.74843655192112560892762487797, −3.513587596604289056387872375, −2.69919382214382219913002947477, −1.37107584917557868149142516617, 1.36911687521153382016760923474, 2.85943766736053804936135673710, 4.22920246892969095132407116237, 5.86600491509152947450737871615, 7.441940271623839510736708500887, 7.86832687736034986991681519078, 9.14466699059157236973151013155, 10.20240467565008705406921613794, 10.64599016511748547938039844615, 12.48270472735137580697377883985, 13.78037766417594530135648682334, 14.68903978771746276325946783788, 15.7085293951560054780631789908, 16.43857418229015831655098733206, 17.643275778037613643455197601831, 18.59523658464725799958806665306, 19.631241278609131222020321482944, 20.46735762668799380385713091909, 20.901828633252200107543933685, 22.680402380157763722099124926611, 23.68973683187977212463155529743, 24.752135346759654224203624308622, 25.77502878864872574373225275977, 26.13607545603345705221809514059, 27.1354201136030452878994500664

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