L-function 1-967-967.138-r0-0-0

• Label: 1-967-967.138-r0-0-0
• Degree: $1$
• Conductor: $967$
• Sign: $-0.476 + 0.878i$
• Analytic cond.: $4.49072$
• Root an. cond.: $4.49072$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.929 + 0.368i)2^-s + (0.924 + 0.380i)3^-s + (0.728 + 0.684i)4^-s + (−0.906 + 0.422i)5^-s + (0.719 + 0.694i)6^-s + (0.602 − 0.797i)7^-s + (0.425 + 0.905i)8^-s + (0.710 + 0.703i)9^-s + (−0.998 + 0.0585i)10^-s + (−0.844 + 0.536i)11^-s + (0.413 + 0.910i)12^-s + (−0.0422 + 0.999i)13^-s + (0.854 − 0.519i)14^-s + (−0.998 + 0.0455i)15^-s + (0.0617 + 0.998i)16^-s + (−0.967 + 0.250i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$967$
Sign:	$-0.476 + 0.878i$
Analytic conductor:	$4.49072$
Root analytic conductor:	$4.49072$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{967} (138, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 967,\ (0:\ ),\ -0.476 + 0.878i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.564071614 + 2.628281433i$
$L(1)$	$\approx$	$1.762353718 + 1.136941652i$

Euler product

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

	$p$	$F_p(T)$
bad	967	$1$
good	2	$1 + (0.929 + 0.368i)T$
	3	$1 + (0.924 + 0.380i)T$
	5	$1 + (-0.906 + 0.422i)T$
	7	$1 + (0.602 - 0.797i)T$
	11	$1 + (-0.844 + 0.536i)T$
	13	$1 + (-0.0422 + 0.999i)T$
	17	$1 + (-0.967 + 0.250i)T$
	19	$1 + (-0.971 - 0.238i)T$
	23	$1 + (0.987 + 0.155i)T$

Imaginary part of the first few zeros on the critical line

−21.16361103347928617267800853900, −20.89257086000358661890261778832, −19.92924809530858938689407656152, −19.43036412426259883418920541809, −18.6282413916554955349095348830, −17.87882888071924561491625241608, −16.34304294859541331985335855180, −15.48048885727970141505681297287, −15.15534174889607857518199599030, −14.405028668537575057816582875329, −13.27360822805900746197243052428, −12.82522008626091840737071713896, −12.139201751445263954360991211056, −11.16269317833956275008263410332, −10.485296142854138054709235665483, −9.06191713407473154753659016886, −8.385201145321614348219596427959, −7.6469592819044489007472340535, −6.65954334590695258915357322872, −5.460916052870420149496300304173, −4.71695908231319170913316181232, −3.73835148911375792597218909653, −2.82437937405419666386574064795, −2.162209225378881242044622906262, −0.82405766172420529360157798547, 1.87163963568221695310945101010, 2.72309367006428074107476996830, 3.736992546563818351900749489008, 4.48314184806245921767537150811, 4.85659572180958991255448139485, 6.719372836103315645742002694140, 7.11694122021556848014430487371, 8.07342135650547030066544395691, 8.579499441847178137797575248148, 10.0374777435224996244102372314, 11.00215037015023213901478389975, 11.41323381355860146965007982117, 12.80321531397446092705369221311, 13.30525507466298416928425992586, 14.26239765434697256465149764795, 14.90226527462097083884903527745, 15.33587077949298566003903564591, 16.22348645716461276710614569583, 16.94938727692901578807917584498, 18.10039750953505356612528264126, 19.14461082900015013853608183520, 20.00425686798077672095486320201, 20.37220147153394871050267580, 21.44594633868727455448075930561, 21.723545443617157052648593828361

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