L-function 1-920-920.443-r0-0-0

• Label: 1-920-920.443-r0-0-0
• Degree: $1$
• Conductor: $920$
• Sign: $-0.946 - 0.322i$
• Analytic cond.: $4.27246$
• Root an. cond.: $4.27246$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.540 + 0.841i)3^-s + (0.281 − 0.959i)7^-s + (−0.415 − 0.909i)9^-s + (−0.654 + 0.755i)11^-s + (0.281 + 0.959i)13^-s + (−0.989 + 0.142i)17^-s + (0.142 − 0.989i)19^-s + (0.654 + 0.755i)21^-s + (0.989 + 0.142i)27^-s + (−0.142 − 0.989i)29^-s + (−0.841 + 0.540i)31^-s + (−0.281 − 0.959i)33^-s + (−0.909 + 0.415i)37^-s + (−0.959 − 0.281i)39^-s + (0.415 − 0.909i)41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign:	$-0.946 - 0.322i$
Analytic conductor:	\(4.27246\)
Root analytic conductor:	\(4.27246\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{920} (443, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 920,\ (0:\ ),\ -0.946 - 0.322i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.01925933157 + 0.1162780571i\)
\(L(1)\)	\(\approx\)	\(0.6614349256 + 0.1504189261i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	5	\( 1 \)
	23	\( 1 \)
good	3	\( 1 + (-0.540 + 0.841i)T \)
	7	\( 1 + (0.281 - 0.959i)T \)
	11	\( 1 + (-0.654 + 0.755i)T \)
	13	\( 1 + (0.281 + 0.959i)T \)
	17	\( 1 + (-0.989 + 0.142i)T \)
	19	\( 1 + (0.142 - 0.989i)T \)
	29	\( 1 + (-0.142 - 0.989i)T \)
	31	\( 1 + (-0.841 + 0.540i)T \)
	37	\( 1 + (-0.909 + 0.415i)T \)

Imaginary part of the first few zeros on the critical line

−21.616659008476263693794473653903, −20.6358778643275227402206988250, −19.77380896231372675458579441237, −18.81475152813908265377944432829, −18.24739668940175293699199650465, −17.813451891331331382418187035387, −16.68859249724192677648618080090, −15.97219278966946817465062005015, −15.117429063487707228783800787, −14.13872535445517089729231495828, −13.183583808559712539168876712617, −12.68076749310228904968232805723, −11.76176470443363929995490768474, −11.04608840994808008703986368318, −10.31007003902028697474191060949, −8.88149522854583095660689611229, −8.26793573054999539413973278834, −7.463289044057706967043075033329, −6.35873058644364465857136975321, −5.584447630913090145075700543673, −5.08066243670714452040140096942, −3.45858811151772484171351945308, −2.44668745424141369929630533908, −1.534906121216593139321319007295, −0.05407182020441760708324404006, 1.52869199404370709725464733401, 2.847399789673222921786634113646, 4.20824375441440412692666547350, 4.460251827686913799827744239766, 5.514020435882123925290945718726, 6.67836937076313076541724323971, 7.28692632045488734605465940885, 8.57602171435316148769856285102, 9.414522793998619303604466732248, 10.23070860365553055984715947280, 11.00566473860248582509204777289, 11.52346587679170080819021050709, 12.67030543336244453827249576047, 13.56991940299268803733512723476, 14.38795459115125091285763217731, 15.36830407608425180527388611055, 15.907004160069768921947216348233, 16.811765295569777990049233473364, 17.52720153678945588320850046763, 18.03469898402652110387583739284, 19.29279129936914984072550460751, 20.19882615373073247516097129986, 20.768424179092351550800527980521, 21.48883212698520566176516160114, 22.32625941819059409254572171460

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