L-function 1-1360-1360.659-r1-0-0

• Label: 1-1360-1360.659-r1-0-0
• Degree: $1$
• Conductor: $1360$
• Sign: $-0.963 + 0.266i$
• Analytic cond.: $146.152$
• Root an. cond.: $146.152$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− 3^-s + i·7^-s + 9^-s + 11^-s − i·13^-s + i·19^-s − i·21^-s + i·23^-s − 27^-s − 29^-s − i·31^-s − 33^-s − 37^-s + i·39^-s + i·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(1360\)    =    \(2^{4} \cdot 5 \cdot 17\)
Sign:	$-0.963 + 0.266i$
Analytic conductor:	\(146.152\)
Root analytic conductor:	\(146.152\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1360} (659, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 1360,\ (1:\ ),\ -0.963 + 0.266i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.08405658274 + 0.6183679294i\)
\(L(1)\)	\(\approx\)	\(0.7477371673 + 0.1464721586i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	5	\( 1 \)
	17	\( 1 \)
good	3	\( 1 + T \)
	7	\( 1 - T \)
	11	\( 1 \)
	13	\( 1 \)
	19	\( 1 + iT \)
	23	\( 1 \)
	29	\( 1 + T \)
	31	\( 1 \)
	37	\( 1 + T \)

Imaginary part of the first few zeros on the critical line

−20.39371649640008952906809446611, −19.47349816244039023184654629695, −18.84581044423809978905472350713, −17.88131239218919812896092054951, −17.12923982390257449893419692203, −16.75772998564995930983578974335, −16.00857171057136956916709847169, −15.056655948629647400099838455133, −14.04346863095601667040296702388, −13.52967405687983861715746281523, −12.40016912165889962313032566759, −11.88639616845631620909349337567, −10.9123527797488502240202480262, −10.54144947557375948940683283028, −9.43371993811249463315690608623, −8.76686424660257544683265635766, −7.26030422467119563093668247606, −6.950734243686733170120780613733, −6.15763681400726344954378006895, −5.03577173236534382521549888513, −4.28856330260471560317973088187, −3.63237953362181984594993596796, −2.02034747932434276028311013627, −1.07937500710177734000177213188, −0.16446851564869519159451285089, 1.1145820594972383450107154210, 1.996111893358711722511027070479, 3.33943906087227802463880019085, 4.19390961600708952753446599193, 5.43121747558313990942937780081, 5.725762541796387329743477069917, 6.623888374663310202166986432711, 7.598565338367192352195436508639, 8.488654945746655079164839747001, 9.578921833610779078735047531467, 10.04629237478364249083721832529, 11.355863964132517393741890253854, 11.55093496882238080200884468401, 12.58989695085189441071499946842, 12.99138837479618121964819475874, 14.27402974611647762742030995719, 15.102308103844909007920237954938, 15.706138967248233824810469961933, 16.56464269749125222222779512161, 17.28868189324764582585800164208, 17.90949301972257756594141478088, 18.704757724719719731049475572326, 19.28788749833761750180197490242, 20.37063166745201981014197897726, 21.15825477671290505053705144925

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