L-function 1-837-837.158-r0-0-0

• Label: 1-837-837.158-r0-0-0
• Degree: $1$
• Conductor: $837$
• Sign: $-0.892 + 0.451i$
• Analytic cond.: $3.88701$
• Root an. cond.: $3.88701$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.848 + 0.529i)2^-s + (0.438 − 0.898i)4^-s + (−0.173 − 0.984i)5^-s + (−0.241 − 0.970i)7^-s + (0.104 + 0.994i)8^-s + (0.669 + 0.743i)10^-s + (−0.241 − 0.970i)11^-s + (−0.0348 + 0.999i)13^-s + (0.719 + 0.694i)14^-s + (−0.615 − 0.788i)16^-s + (−0.809 + 0.587i)17^-s + (−0.978 + 0.207i)19^-s + (−0.961 − 0.275i)20^-s + (0.719 + 0.694i)22^-s + (−0.719 − 0.694i)23^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$837$    =    $3^{3} \cdot 31$
Sign:	$-0.892 + 0.451i$
Analytic conductor:	$3.88701$
Root analytic conductor:	$3.88701$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{837} (158, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 837,\ (0:\ ),\ -0.892 + 0.451i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.01320357563 - 0.05534814102i$
$L(1)$	$\approx$	$0.5145006917 - 0.06716390970i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	31	$1$
good	2	$1 + (-0.848 + 0.529i)T$
	5	$1 + (-0.173 - 0.984i)T$
	7	$1 + (-0.241 - 0.970i)T$
	11	$1 + (-0.241 - 0.970i)T$
	13	$1 + (-0.0348 + 0.999i)T$
	17	$1 + (-0.809 + 0.587i)T$
	19	$1 + (-0.978 + 0.207i)T$
	23	$1 + (-0.719 - 0.694i)T$
	29	$1 + (0.848 - 0.529i)T$

Imaginary part of the first few zeros on the critical line

−22.464297348667512822180238502141, −21.737643669223398667248692964422, −20.99866163554544100341904313977, −19.873741623184091585287189830802, −19.507134702211754560703948071028, −18.480777452541181858929138283929, −17.91478360784701619894076851349, −17.497481880432273764120220114181, −16.03262182766665340192127578176, −15.47661936913240241315583589659, −14.85383999522331194240028695076, −13.495236608109418990234858293647, −12.5513688998137817450194684347, −11.94280534667594344536964107242, −10.98801986887121132660550744401, −10.30169652161681537759587105751, −9.542290328820200836839582699822, −8.60859666566960486560871203136, −7.72413794615767060388004658939, −6.90906177217075606634950044923, −6.06320315030348964145511622932, −4.676655272992044921674490299459, −3.40425152971745331065741739642, −2.603133739552966828319757913447, −1.92699600627733946459874146097, 0.03556352940731866620592175108, 1.166009831083655647227985579041, 2.25907636675544369895028009909, 3.973901681442472920288294385270, 4.65815609417460631185709780873, 5.96417143598963849084489205330, 6.58424695128921777209031321130, 7.66852003288424891237097630744, 8.5071279243167779204508198173, 8.99105643449363683699987285111, 10.119314531333021171134545283, 10.794055553493702552061231557474, 11.69529116937705029199867480593, 12.80729488622099737184391020739, 13.74757742842049151914353152243, 14.37358694428547353867614777352, 15.69267830537105717309653942981, 16.15354607094195185202468463157, 16.95399552462713206527867519121, 17.33588104127579180407045598984, 18.50631877763605254935628554997, 19.54760173822077480400895827910, 19.62564654927164749445663366095, 20.82991564564129741769009528563, 21.35118933578023495473917306961

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