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Properties

Label 1-185-185.32-r0-0-0
Degree $1$
Conductor $185$
Sign $-0.826 + 0.562i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.826 + 0.562i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ -0.826 + 0.562i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1350681204 + 0.4385898302i\)
\(L(\frac12)\) \(\approx\) \(0.1350681204 + 0.4385898302i\)
\(L(1)\) \(\approx\) \(0.4498315098 + 0.3071440280i\)
\(L(1)\) \(\approx\) \(0.4498315098 + 0.3071440280i\)

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

−27.25629201990888399740875663961, −25.755400703958729863003486646395, −25.03658383273606949294552137000, −24.222926112977631819244499346374, −22.65573993510715181408943641279, −22.13513964015593724386963792452, −21.05749067349664606350443518234, −19.68876193812976343813713701516, −19.03435642095950345962380840544, −18.24012608883008720453392326131, −17.34440842979340408082090078070, −16.46263808495413946278116705439, −15.345655235139904275595381315334, −13.49595893492746883926537186578, −12.76137671453691455744104878524, −11.680432005819534718175382050362, −11.13778963012733671834111841623, −9.722590999521310786464631499373, −8.66166428289824138310209281671, −7.60423665915141077176481640282, −6.432067240168888911632196437731, −5.25008201944923850057429059910, −3.25268214355894082551171918337, −2.1326024379572245341172353245, −0.52416559554777705183000583370, 1.51536009923940279388611322095, 3.90174590450880926515967166130, 4.890445376011095280784689302628, 6.36963200126545152373413595085, 6.976474918562864269582431710644, 8.525409266785704659694353408545, 9.68489095066983062621031192352, 10.32426641573246810258176645965, 11.302013385816020068710472822001, 12.66183973749899681927560969372, 14.39142221192436446830447346900, 14.94528769160590265841373694204, 16.31952213731308231626185283952, 16.81322882214950317973066327502, 17.52197725515038023627556097778, 18.77350821427819018486592253531, 19.85465663737484676359669416829, 20.72717331578548564544055325695, 21.99912415199571363588016790426, 23.24113983926989735797332869219, 23.49863870850240758253692942936, 24.897357204723377659748398723427, 26.05558005675800407051516389631, 26.578630579064652181469825086875, 27.511729977464820765572074968120

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