L-function 1-812-812.499-r1-0-0

• Label: 1-812-812.499-r1-0-0
• Degree: $1$
• Conductor: $812$
• Sign: $0.954 - 0.297i$
• Analytic cond.: $87.2615$
• Root an. cond.: $87.2615$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.826 − 0.563i)3^-s + (−0.733 + 0.680i)5^-s + (0.365 − 0.930i)9^-s + (0.365 + 0.930i)11^-s + (0.623 − 0.781i)13^-s + (−0.222 + 0.974i)15^-s + (0.5 − 0.866i)17^-s + (0.826 + 0.563i)19^-s + (−0.955 + 0.294i)23^-s + (0.0747 − 0.997i)25^-s + (−0.222 − 0.974i)27^-s + (0.955 + 0.294i)31^-s + (0.826 + 0.563i)33^-s + (−0.365 + 0.930i)37^-s + (0.0747 − 0.997i)39^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$812$    =    $2^{2} \cdot 7 \cdot 29$
Sign:	$0.954 - 0.297i$
Analytic conductor:	$87.2615$
Root analytic conductor:	$87.2615$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{812} (499, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 812,\ (1:\ ),\ 0.954 - 0.297i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.726547545 - 0.4147982639i$
$L(1)$	$\approx$	$1.389366911 - 0.1297510615i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	7	$1$
	29	$1$
good	3	$1 + (0.826 - 0.563i)T$
	5	$1 + (-0.733 + 0.680i)T$
	11	$1 + (0.365 + 0.930i)T$
	13	$1 + (0.623 - 0.781i)T$
	17	$1 + (0.5 - 0.866i)T$
	19	$1 + (0.826 + 0.563i)T$
	23	$1 + (-0.955 + 0.294i)T$
	31	$1 + (0.955 + 0.294i)T$
	37	$1 + (-0.365 + 0.930i)T$

Imaginary part of the first few zeros on the critical line

−21.86587984861963865332594839827, −21.156575861197836374925507628810, −20.51892257140349797036893994675, −19.54450610472888516514435326224, −19.27118621858261309689931265739, −18.24580710644913079974657333189, −16.86088385774775593734465769311, −16.32804004470722078326176694749, −15.685901770809493621688839787452, −14.84448156046016310095889233299, −13.874730198332575062470189005921, −13.37857670449376956727966472873, −12.149807804489142525100743785188, −11.439591733900581070532484162711, −10.47110899193200014264522695355, −9.470037473400824048255609494390, −8.60357100479400321634514511025, −8.24486900573394730958789489836, −7.15582523325989211672234061788, −5.90025392504481826444585671499, −4.83312926533137042298968957708, −3.866480477588496926452703722074, −3.42045676557138154394556492886, −1.98096251130142595595409509013, −0.812529138021652058626242530540, 0.775231516543160490828359536146, 1.91715896736592926078089560612, 3.12417312305440307980595321507, 3.59401536914996353251529231094, 4.82776114654238516346829317172, 6.23492370986691074818535571075, 7.03452034317596677388981649101, 7.823617134747396485410335593363, 8.35949103306071637429090908039, 9.6838495962995398098586087620, 10.20169303164633159532856176039, 11.66813891786236382102360711692, 12.0279718291459754206675207694, 13.092155111346840538806568905341, 14.00666304774486857926457641071, 14.599326730564468220755704779894, 15.47035662020913656481763417988, 16.00977829572899664576199788330, 17.44155179919502326308911130538, 18.28691948926630720000477867207, 18.648478230660666733814832027868, 19.74489405778603086239741842226, 20.19940586953527790841148971153, 20.89579007652447330922431130910, 22.14716998000314920373055170496

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