L-function 1-1800-1800.1067-r1-0-0

• Label: 1-1800-1800.1067-r1-0-0
• Degree: $1$
• Conductor: $1800$
• Sign: $-0.959 + 0.282i$
• Analytic cond.: $193.436$
• Root an. cond.: $193.436$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.866 + 0.5i)7^-s + (0.104 + 0.994i)11^-s + (−0.994 − 0.104i)13^-s + (0.951 − 0.309i)17^-s + (−0.309 − 0.951i)19^-s + (0.406 − 0.913i)23^-s + (−0.669 − 0.743i)29^-s + (−0.669 + 0.743i)31^-s + (−0.587 + 0.809i)37^-s + (0.104 − 0.994i)41^-s + (0.866 + 0.5i)43^-s + (−0.743 + 0.669i)47^-s + (0.5 + 0.866i)49^-s + (−0.951 − 0.309i)53^-s + (−0.104 + 0.994i)59^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1800$    =    $2^{3} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.959 + 0.282i$
Analytic conductor:	$193.436$
Root analytic conductor:	$193.436$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1800} (1067, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1800,\ (1:\ ),\ -0.959 + 0.282i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.09646409592 + 0.6694144505i$
$L(1)$	$\approx$	$0.9963053315 + 0.1293904951i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	5	$1$
good	7	$1 + (0.866 + 0.5i)T$
	11	$1 + (0.104 + 0.994i)T$
	13	$1 + (-0.994 - 0.104i)T$
	17	$1 + (0.951 - 0.309i)T$
	19	$1 + (-0.309 - 0.951i)T$
	23	$1 + (0.406 - 0.913i)T$
	29	$1 + (-0.669 - 0.743i)T$
	31	$1 + (-0.669 + 0.743i)T$
	37	$1 + (-0.587 + 0.809i)T$

Imaginary part of the first few zeros on the critical line

−19.63795915891754131362181441409, −18.94463227461662045684319991678, −18.299135961113766136295720726995, −17.23579831715911714484814552567, −16.89933999895162409038437594603, −16.15648860337990787472185163541, −15.042675450617810004969312586567, −14.42205211464508986563079245064, −13.971020588106939275169128542779, −12.92200278498177176062532692626, −12.222230791212589483686779577677, −11.29225858428952970754005168351, −10.81844652459212540695948542736, −9.86364486524769343630465121491, −9.122980109026974972606774002706, −8.00500206945744051893304005431, −7.70721137305589450586070460448, −6.68408288443592026951290131596, −5.59316254514869232344902787903, −5.10907234759745571323459631469, −3.92382074682955305065020894687, −3.34694083988633917431251237547, −2.03707793681184559013034986755, −1.26162682446729136168163639776, −0.1198147962112678112529047198, 1.18242551532233775309464952902, 2.18991819116394229043579579678, 2.8415235047878889619821854539, 4.18621265489692030399927881285, 4.918572221005869702788495274453, 5.475322045239232401138889565415, 6.71061606326115314560570571105, 7.400746388951366934973046605067, 8.127119457976757570645384260, 9.08059426305486129171769971635, 9.71762445885593807998968657556, 10.62154100608486758736905508879, 11.40358413939689370803633781534, 12.28508297959659829902311440498, 12.61801373986482230012582770260, 13.79210291043536233289703040677, 14.68083835374295402270859869297, 14.93554368581356664826296258416, 15.82845675740781405129388560544, 16.837007758393416816445997517416, 17.46999585647503874085778952093, 18.01773122201554723061525967103, 18.91946769263984269811519270067, 19.53789006738545293074078454366, 20.546334898230722554242230846486

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