L-function 1-5200-5200.1653-r0-0-0

• Label: 1-5200-5200.1653-r0-0-0
• Degree: $1$
• Conductor: $5200$
• Sign: $-0.989 - 0.146i$
• Analytic cond.: $24.1486$
• Root an. cond.: $24.1486$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.978 − 0.207i)3^-s + (0.5 + 0.866i)7^-s + (0.913 + 0.406i)9^-s + (−0.913 + 0.406i)11^-s + (−0.207 − 0.978i)17^-s + (−0.978 + 0.207i)19^-s + (−0.309 − 0.951i)21^-s + (0.406 + 0.913i)23^-s + (−0.809 − 0.587i)27^-s + (0.207 − 0.978i)29^-s + (−0.951 − 0.309i)31^-s + (0.978 − 0.207i)33^-s + (0.994 − 0.104i)37^-s + (0.994 − 0.104i)41^-s + (−0.5 − 0.866i)43^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign:	$-0.989 - 0.146i$
Analytic conductor:	\(24.1486\)
Root analytic conductor:	\(24.1486\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5200} (1653, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 5200,\ (0:\ ),\ -0.989 - 0.146i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.007681627892 + 0.1043254429i\)
\(L(1)\)	\(\approx\)	\(0.6754825690 + 0.05487247998i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	5	\( 1 \)
	13	\( 1 \)
good	3	\( 1 + (-0.978 - 0.207i)T \)
	7	\( 1 + (0.5 + 0.866i)T \)
	11	\( 1 + (-0.913 + 0.406i)T \)
	17	\( 1 + (-0.207 - 0.978i)T \)
	19	\( 1 + (-0.978 + 0.207i)T \)
	23	\( 1 + (0.406 + 0.913i)T \)
	29	\( 1 + (0.207 - 0.978i)T \)
	31	\( 1 + (-0.951 - 0.309i)T \)
	37	\( 1 + (0.994 - 0.104i)T \)

Imaginary part of the first few zeros on the critical line

−17.60773357361195305050355771983, −16.85491458131996202309817196604, −16.51994698255796606059757752396, −15.80380622748094953490231890266, −14.92514802977841401799316002437, −14.51759640763698916521291879899, −13.44567100445923009970118259114, −12.83842735439507285207819284534, −12.469246838354099924897604368, −11.24324687159188902856385237942, −10.95104547737475030476937332029, −10.48102664605752924618963623402, −9.794844786499058938258180512605, −8.71189215509040709090473185459, −8.13225962948512015273205555682, −7.246345125975435074031095336529, −6.66128473984145299904780632817, −5.8990795642308712430175718812, −5.1801883593575646135627899203, −4.456624458808123025969873070, −3.96485676246461365198223449671, −2.9177976748840038426629174372, −1.850871146833012641925235490411, −1.00946639283755721405669623021, −0.036950214983004701421446682436, 1.12699112710028730362245229207, 2.174528606442294236492023495409, 2.55719013075399502172221875946, 3.9298817491200608919585998290, 4.68293919526384930281265068602, 5.34483570079048710336790855909, 5.800986347269517632896438334686, 6.63919263202185653168178363091, 7.486036603029003573938483418845, 7.92704696653850730784485715121, 8.92611641434809676901562981276, 9.63369662330157134807231873346, 10.364511981271494179317999819188, 11.15181396409322882097747697750, 11.56522562065443671041449708308, 12.24223598162708224233084171746, 13.02122316426999151327871168621, 13.33288953441817330271392439056, 14.51565623455744399040623245602, 15.08515187417884783146871067099, 15.86126703163479024068427817730, 16.17784527520155517329087810740, 17.2386325349800600100768906821, 17.64859802555400645755463928463, 18.2389984715233422285283714139

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