L-function 1-185-185.128-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	37	$1$
good	2	$1 + (0.939 - 0.342i)T$
	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	$1 + (-0.866 - 0.5i)T$

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