L-function 1-185-185.28-r1-0-0

• Label: 1-185-185.28-r1-0-0
• Degree: $1$
• Conductor: $185$
• Sign: $0.123 + 0.992i$
• Analytic cond.: $19.8810$
• Root an. cond.: $19.8810$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.342 − 0.939i)2^-s + (0.342 − 0.939i)3^-s + (−0.766 + 0.642i)4^-s − 6^-s + (0.984 − 0.173i)7^-s + (0.866 + 0.5i)8^-s + (−0.766 − 0.642i)9^-s + (−0.5 + 0.866i)11^-s + (0.342 + 0.939i)12^-s + (−0.642 − 0.766i)13^-s + (−0.5 − 0.866i)14^-s + (0.173 − 0.984i)16^-s + (−0.642 + 0.766i)17^-s + (−0.342 + 0.939i)18^-s + (−0.939 − 0.342i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$185$    =    $5 \cdot 37$
Sign:	$0.123 + 0.992i$
Analytic conductor:	$19.8810$
Root analytic conductor:	$19.8810$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{185} (28, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 185,\ (1:\ ),\ 0.123 + 0.992i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.07764230224 - 0.06860494265i$
$L(1)$	$\approx$	$0.5571864698 - 0.4649598910i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	37	$1$
good	2	$1 + (-0.342 - 0.939i)T$
	3	$1 + (0.342 - 0.939i)T$
	7	$1 + (0.984 - 0.173i)T$
	11	$1 + (-0.5 + 0.866i)T$
	13	$1 + (-0.642 - 0.766i)T$
	17	$1 + (-0.642 + 0.766i)T$
	19	$1 + (-0.939 - 0.342i)T$
	23	$1 + (-0.866 + 0.5i)T$
	29	$1 + (-0.5 + 0.866i)T$

Imaginary part of the first few zeros on the critical line

−27.44505377989606637957861343253, −26.64312341308505415770697002715, −26.09420951887499598213499730852, −24.77960319759289350905819340938, −24.21727618032826405271136521370, −23.04842583702382690413121209753, −21.89603192446310984198663334522, −21.20236964448163113905483255021, −19.965052041371355587562034775236, −18.87423229316525290456522785024, −17.87365857203155987277261175257, −16.72986225655454194320714384146, −16.11349794805524063761345798225, −14.9518261849594485592761317670, −14.38631610465511823750567047829, −13.402889749249371173603900740044, −11.48492527072691791605669242713, −10.527241812055065680933652463, −9.37252113011787903264069931821, −8.5061122625855313690365346701, −7.66173626601189153833591557266, −6.070829476952714062960550613464, −4.96215037539727804920190935784, −4.138270476235421614399393211964, −2.20889559731793844957543253956, 0.03697051673331892450218130021, 1.669144028536447424141469937048, 2.40594213464595406201162837590, 3.98931185231301669687214768492, 5.332267213685186994191686999991, 7.22420489258227128220059148050, 7.99893795333699079509320912217, 8.94148607992179937945311470915, 10.33114440391537706940953944152, 11.27146834675853157344908285592, 12.46772644291451607070688138588, 12.99179471036084348245722127264, 14.21623998709630585158355337405, 15.131955355420129546410089610563, 17.168666484935711475317333510795, 17.72729366182788137398357350384, 18.44474982642291088982883399296, 19.72377965129383091205344990720, 20.16331854162402436562955930507, 21.19107143902517803989522693210, 22.298488167003427210303528726664, 23.473266404618123370702542075130, 24.205256209682418257155383681934, 25.53792130913442066714358848746, 26.1868408933931006962507669077

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