L-function 1-3895-3895.1039-r0-0-0

• Label: 1-3895-3895.1039-r0-0-0
• Degree: $1$
• Conductor: $3895$
• Sign: $-0.329 + 0.943i$
• Analytic cond.: $18.0883$
• Root an. cond.: $18.0883$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.984 + 0.173i)2^-s + (−0.996 + 0.0871i)3^-s + (0.939 + 0.342i)4^-s + (−0.996 − 0.0871i)6^-s + (−0.965 − 0.258i)7^-s + (0.866 + 0.5i)8^-s + (0.984 − 0.173i)9^-s + (0.258 + 0.965i)11^-s + (−0.965 − 0.258i)12^-s + (−0.0871 + 0.996i)13^-s + (−0.906 − 0.422i)14^-s + (0.766 + 0.642i)16^-s + (0.819 − 0.573i)17^-s + 18^-s + (0.984 + 0.173i)21^-s + (0.0871 + 0.996i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3895$    =    $5 \cdot 19 \cdot 41$
Sign:	$-0.329 + 0.943i$
Analytic conductor:	$18.0883$
Root analytic conductor:	$18.0883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3895} (1039, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3895,\ (0:\ ),\ -0.329 + 0.943i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.129860160 + 1.591832876i$
$L(1)$	$\approx$	$1.274112556 + 0.4397049392i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	19	$1$
	41	$1$
good	2	$1 + (0.984 + 0.173i)T$
	3	$1 + (-0.996 + 0.0871i)T$
	7	$1 + (-0.965 - 0.258i)T$
	11	$1 + (0.258 + 0.965i)T$
	13	$1 + (-0.0871 + 0.996i)T$
	17	$1 + (0.819 - 0.573i)T$
	23	$1 + (-0.939 - 0.342i)T$
	29	$1 + (0.819 + 0.573i)T$
	31	$1 + (-0.5 - 0.866i)T$

Imaginary part of the first few zeros on the critical line

−18.42348489891695299561180919785, −17.52126648086949244728931197688, −16.764109212995452555619314480729, −16.11967224637837067874884871509, −15.77913842348929675194461362649, −14.95225705554681272779637866409, −14.10028302237421251879806616490, −13.333472310713104944074654272255, −12.78033357600207376205338911218, −12.204037021766445784960169256716, −11.647995531398399164489695900506, −10.8312187926538295614400023015, −10.19979546509961501111771399059, −9.719314468592042714840751023554, −8.38759704296688095961894172147, −7.57300632144623769606570005509, −6.69442644745564647885685976493, −6.03528539060961747692726347, −5.696334013757921730888027716052, −4.96303545072407511910163125021, −3.85086558571430011959686726279, −3.41823070518766458060504245754, −2.4890060931416627449288337452, −1.3833349963355162231896546630, −0.48021219871676943964445674292, 1.102276211715809681368935987272, 2.052217487759799533794876356312, 3.0185657633624030445516091, 4.00319913205991818277408507920, 4.4485013795247377008583296431, 5.18328210047180560575826005716, 6.22829645716331946939349355677, 6.40147847042562620505497549104, 7.27287369736384883754028081235, 7.790261078181931131946748950088, 9.335289861796386178660982509057, 9.83283360286081857886165578228, 10.55721804266191219152108620347, 11.45093665073677213982255238554, 11.94110305221007838902260335182, 12.64009671772186712047058087951, 12.99044236190694885952135967487, 14.071186890372092711322035205904, 14.49007909743139013232049290017, 15.51321740469931754367928687535, 16.03693596402500711173555041820, 16.611168934025440510050060028540, 17.02094028570522519804956417902, 17.97072938991197842085553306336, 18.689465710803662630795509128451

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