L-function 1-1800-1800.83-r1-0-0

• Label: 1-1800-1800.83-r1-0-0
• Degree: $1$
• Conductor: $1800$
• Sign: $-0.916 - 0.400i$
• 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 + (−0.913 − 0.406i)11^-s + (−0.406 − 0.913i)13^-s + (−0.951 + 0.309i)17^-s + (−0.309 − 0.951i)19^-s + (0.994 − 0.104i)23^-s + (0.978 − 0.207i)29^-s + (0.978 + 0.207i)31^-s + (0.587 − 0.809i)37^-s + (−0.913 + 0.406i)41^-s + (0.866 − 0.5i)43^-s + (0.207 + 0.978i)47^-s + (0.5 − 0.866i)49^-s + (0.951 + 0.309i)53^-s + (0.913 − 0.406i)59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign:	$-0.916 - 0.400i$
Analytic conductor:	\(193.436\)
Root analytic conductor:	\(193.436\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1800} (83, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 1800,\ (1:\ ),\ -0.916 - 0.400i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.2680508032 - 1.283086824i\)
\(L(1)\)	\(\approx\)	\(0.9874937533 - 0.2705301182i\)

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	\( 1 + (-0.913 - 0.406i)T \)
	13	\( 1 + (-0.406 - 0.913i)T \)
	17	\( 1 + (-0.951 + 0.309i)T \)
	19	\( 1 + (-0.309 - 0.951i)T \)
	23	\( 1 + (0.994 - 0.104i)T \)
	29	\( 1 + (0.978 - 0.207i)T \)
	31	\( 1 + (0.978 + 0.207i)T \)
	37	\( 1 + (0.587 - 0.809i)T \)

Imaginary part of the first few zeros on the critical line

−20.448080517200606087274243687446, −19.572367398495224965524704739392, −18.69523105760995763862140736622, −18.2455512404763494490929383355, −17.397921701445443138737723800550, −16.76706173516850851079517618527, −15.74828011982858354314932604119, −15.20054496707738660490057398979, −14.46519121516187844131933285241, −13.689879696724751074533916896, −12.89285511162796880698318798341, −11.98569866870057613050957861073, −11.486666152578176985481656063937, −10.55377545968032416196678522632, −9.85273515938749509341782774070, −8.81749707692406092108084269996, −8.305024963054103772113525105351, −7.35602238421045823845908187903, −6.625432736905731094005452275877, −5.568715023993554883731455743526, −4.782887174776265458547690786725, −4.23258126969288689639953817048, −2.78329025572437077490530842127, −2.19140935990134184217383749960, −1.18955853643947675277386797014, 0.25367814957670586375968750836, 1.06376393676510245599160402146, 2.385440440749636231774032551279, 2.97082007624397638190430244826, 4.316223758960526334810530600957, 4.85562847115206517773984352005, 5.698062228304393311974519540848, 6.755776148188261367872773111794, 7.53209790498262226227636384311, 8.28653403510872881707268201274, 8.899546675451393148738644732848, 10.10247756647609440894921532538, 10.75896675225957031057601201272, 11.21633650365099853236631505842, 12.2710619382698482089190981332, 13.164657390291053881432322840457, 13.59472122835415236836448753526, 14.57760006345246200817340993835, 15.3178361365883364191513059287, 15.825224770114769769319146863383, 16.94051744578271007078626535168, 17.596115869731354186953090871463, 17.97463160225124048424047939139, 19.03613080265764749076059250466, 19.73691295269342892981256396188

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