L-function 1-1045-1045.569-r0-0-0

• Label: 1-1045-1045.569-r0-0-0
• Degree: $1$
• Conductor: $1045$
• Sign: $0.970 - 0.242i$
• Analytic cond.: $4.85295$
• Root an. cond.: $4.85295$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.309 + 0.951i)2^-s + (−0.809 + 0.587i)3^-s + (−0.809 − 0.587i)4^-s + (−0.309 − 0.951i)6^-s + (−0.809 − 0.587i)7^-s + (0.809 − 0.587i)8^-s + (0.309 − 0.951i)9^-s + 12^-s + (−0.309 + 0.951i)13^-s + (0.809 − 0.587i)14^-s + (0.309 + 0.951i)16^-s + (0.309 + 0.951i)17^-s + (0.809 + 0.587i)18^-s + 21^-s − 23^-s + (−0.309 + 0.951i)24^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1045$    =    $5 \cdot 11 \cdot 19$
Sign:	$0.970 - 0.242i$
Analytic conductor:	$4.85295$
Root analytic conductor:	$4.85295$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1045} (569, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1045,\ (0:\ ),\ 0.970 - 0.242i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3429665136 - 0.04215966795i$
$L(1)$	$\approx$	$0.4548600805 + 0.2420944032i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	11	$1$
	19	$1$
good	2	$1 + (-0.309 + 0.951i)T$
	3	$1 + (-0.809 + 0.587i)T$
	7	$1 + (-0.809 - 0.587i)T$
	13	$1 + (-0.309 + 0.951i)T$
	17	$1 + (0.309 + 0.951i)T$
	23	$1 - T$
	29	$1 + (-0.809 - 0.587i)T$
	31	$1 + (-0.309 + 0.951i)T$
	37	$1 + (-0.809 - 0.587i)T$

Imaginary part of the first few zeros on the critical line

−21.81968324476723487957311540586, −20.60082141023963195426225471396, −20.01116218849914423279227453810, −18.987843003971627905963679338312, −18.635814574746116731558818700305, −17.86141851775092489574477261132, −17.12376749057658505741369682850, −16.32615380194318577350363266237, −15.511996133886287766074916217625, −14.14909128986421006927475523696, −13.30705915789712285109866401445, −12.61539973378782782820655544278, −12.082045691385163199127014084988, −11.34894368461731142042730790628, −10.38599893540885432645930863123, −9.78705421868169889618770593989, −8.82418021468702921860105418816, −7.78778916841854888412323373585, −7.068935968254565135860171008192, −5.773692298232011161499322739828, −5.25402688146794948079464901503, −3.99403219891259908495918356597, −2.88374406410455142618704620052, −2.12782144789846372403009793372, −0.85526618304564913751494492371, 0.24704737836385679077592904813, 1.66459471064338123885467229207, 3.66933362696989145048028768407, 4.158262511772334790760545061546, 5.23395302163544445132876545387, 6.06844597513559303626355842490, 6.71777203555932320399973222845, 7.48881663220829915117300817279, 8.69677310430059740064954050068, 9.53446169585495453490177437421, 10.15236996110709096596892693335, 10.83224141839716217578873938042, 12.02498679778995092295111560172, 12.84214694636383163963542677970, 13.83098501563859294849138918495, 14.61047527198421177118411483604, 15.48650431643691158748680704933, 16.21580708099084910906324452772, 16.73140338573600012563543865453, 17.32445852628715886251630743405, 18.15623208377893577686566774441, 19.08560711135058406810471938707, 19.71080943257989983986851891282, 20.86343393420598142587566690675, 21.87680307252536513399423759469

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