L-function 1-185-185.32-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	5	$1$
	37	$1$
good	2	$1 + (-0.766 + 0.642i)T$
	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	$1 + (-0.866 + 0.5i)T$

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