L-function 1-30e2-900.31-r1-0-0

• Label: 1-30e2-900.31-r1-0-0
• Degree: $1$
• Conductor: $900$
• Sign: $-0.983 + 0.180i$
• Analytic cond.: $96.7184$
• Root an. cond.: $96.7184$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign:	$-0.983 + 0.180i$
Analytic conductor:	\(96.7184\)
Root analytic conductor:	\(96.7184\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{900} (31, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 900,\ (1:\ ),\ -0.983 + 0.180i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.02709537478 + 0.2977280462i\)
\(L(1)\)	\(\approx\)	\(0.9357684196 + 0.04435221541i\)

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	\( 1 + (0.5 - 0.866i)T \)
	11	\( 1 + (0.104 + 0.994i)T \)
	13	\( 1 + (-0.104 + 0.994i)T \)
	17	\( 1 + (0.309 + 0.951i)T \)
	19	\( 1 + (-0.309 - 0.951i)T \)
	23	\( 1 + (-0.913 - 0.406i)T \)
	29	\( 1 + (0.669 + 0.743i)T \)
	31	\( 1 + (-0.669 + 0.743i)T \)
	37	\( 1 + (-0.809 - 0.587i)T \)

Imaginary part of the first few zeros on the critical line

−21.22144101864739708275116582085, −20.80971423793785400347044837793, −19.737791929791377087836670418289, −18.90995920904947317806403354131, −18.24774531087591448357326704598, −17.52964201076178333838844417729, −16.508819752493929186220290789307, −15.785867885277025731413379820561, −14.99531863623090370344296739186, −14.1559344328199719192455365115, −13.42223619704027665808296577276, −12.27981803508663834869834322379, −11.78207067459120098394952146084, −10.82727148473665747806841790764, −9.94508401228545898070595484990, −8.97228045745394133344077549061, −8.15924670303291983215640388311, −7.51977043825194124270189327188, −5.9832847216153694688435569270, −5.69617949502769800000192668907, −4.549018394325483441010702538998, −3.36144924017592868231314336919, −2.525627753466289289088256489506, −1.33007559031397975210191501823, −0.06238474547713107401633880154, 1.40714576721404328281810806625, 2.15775717112592049015290083656, 3.655493915750070405088948509516, 4.40132291822179605501480412288, 5.18492919640775459165318850060, 6.61516278773037144458538475748, 7.07411681581672614238711583652, 8.1247390373768057525356508913, 8.97335559819266863941832659310, 10.033911888303086296881420588391, 10.64746381156037852716598547643, 11.60244701105398592117520415888, 12.43824693666325746358844738280, 13.26970402676153790250216553836, 14.3313150255865571767439094533, 14.64308489918806983802823746407, 15.82044247026865702874122789313, 16.61024718309381706655697249843, 17.466146771533052938955243878576, 17.91993558949687775528592524390, 19.07776557180979810396228057913, 19.85247055135371654719212197920, 20.41377343619777784346841696263, 21.41923146130273703323605535902, 21.94117905236869863195989287411

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