L-function 1-1045-1045.138-r0-0-0

• Label: 1-1045-1045.138-r0-0-0
• Degree: $1$
• Conductor: $1045$
• Sign: $0.881 - 0.472i$
• 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

+ (−0.970 − 0.241i)2^-s + (0.999 + 0.0348i)3^-s + (0.882 + 0.469i)4^-s + (−0.961 − 0.275i)6^-s + (−0.743 + 0.669i)7^-s + (−0.743 − 0.669i)8^-s + (0.997 + 0.0697i)9^-s + (0.866 + 0.5i)12^-s + (−0.829 − 0.559i)13^-s + (0.882 − 0.469i)14^-s + (0.559 + 0.829i)16^-s + (−0.0697 − 0.997i)17^-s + (−0.951 − 0.309i)18^-s + (−0.766 + 0.642i)21^-s + (0.984 + 0.173i)23^-s + (−0.719 − 0.694i)24^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1045$    =    $5 \cdot 11 \cdot 19$
Sign:	$0.881 - 0.472i$
Analytic conductor:	$4.85295$
Root analytic conductor:	$4.85295$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1045} (138, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1045,\ (0:\ ),\ 0.881 - 0.472i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.216578459 - 0.3058632802i$
$L(1)$	$\approx$	$0.9370760327 - 0.09394750394i$

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	$1 + (-0.970 - 0.241i)T$
	3	$1 + (0.999 + 0.0348i)T$
	7	$1 + (-0.743 + 0.669i)T$
	13	$1 + (-0.829 - 0.559i)T$
	17	$1 + (-0.0697 - 0.997i)T$
	23	$1 + (0.984 + 0.173i)T$
	29	$1 + (0.848 - 0.529i)T$
	31	$1 + (-0.104 - 0.994i)T$
	37	$1 + (-0.951 - 0.309i)T$

Imaginary part of the first few zeros on the critical line

−21.29934682414545729067543891535, −20.588065738353125025454394759083, −19.75315795677709490253408620291, −19.28998056086892031227024653079, −18.822335334672626009434518340989, −17.59901289611131083519006208762, −17.01822862849059231205212403227, −16.08921568180527468250644843590, −15.49980588615162546722844807101, −14.52789281490512366518034900692, −14.020086548422123629788269962765, −12.84342020084371110936761836007, −12.193772747782909200866261088269, −10.7678600616348949403326138362, −10.25815734911500401520365338789, −9.40706288023024523847112812514, −8.77809505128140254228021839619, −7.941147102149283216859777262004, −6.90540788221758334680993021509, −6.73972202379000479975927299786, −5.19158827745601450648690194453, −3.94282887024780562317362813918, −3.00551737669781113607170486403, −2.08125178056078533802988759450, −1.011743471372129271667334986192, 0.77077874629457651782772022200, 2.252141449337822193499878379954, 2.744750031501643161722385011386, 3.55780118411381563121575128373, 4.92568060376941158265027431500, 6.225454899144006989174760020931, 7.1758061435049668139713647907, 7.798116989151002289361445885120, 8.76613737069879450141239764642, 9.42562275094612837939514932197, 9.897611214609347652870895409330, 10.89504418705307142841582677123, 12.02393462619837288134453444300, 12.64026386160615930069809156214, 13.47721736898341534461585621768, 14.59717149391911981506427681984, 15.4748687684297284553924745789, 15.82611596633378437172639009929, 16.85783802361652717745657365032, 17.74285062483507907282664391793, 18.64681319486449513577855584074, 19.13304314098891468606314231296, 19.79339630507878074396164764214, 20.5092666021640514625932212872, 21.22439859679951961573962920934

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