L-function 1-3895-3895.1013-r0-0-0

• Label: 1-3895-3895.1013-r0-0-0
• Degree: $1$
• Conductor: $3895$
• Sign: $-0.260 + 0.965i$
• 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

+ (0.882 + 0.469i)2^-s + (0.996 − 0.0871i)3^-s + (0.559 + 0.829i)4^-s + (0.920 + 0.390i)6^-s + (0.0523 + 0.998i)7^-s + (0.104 + 0.994i)8^-s + (0.984 − 0.173i)9^-s + (0.629 − 0.777i)11^-s + (0.629 + 0.777i)12^-s + (−0.920 − 0.390i)13^-s + (−0.422 + 0.906i)14^-s + (−0.374 + 0.927i)16^-s + (−0.325 + 0.945i)17^-s + (0.951 + 0.309i)18^-s + (0.139 + 0.990i)21^-s + (0.920 − 0.390i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3895$    =    $5 \cdot 19 \cdot 41$
Sign:	$-0.260 + 0.965i$
Analytic conductor:	$18.0883$
Root analytic conductor:	$18.0883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3895} (1013, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3895,\ (0:\ ),\ -0.260 + 0.965i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.828881600 + 3.693619254i$
$L(1)$	$\approx$	$2.203737558 + 1.176274221i$

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	$1 + (0.882 + 0.469i)T$
	3	$1 + (0.996 - 0.0871i)T$
	7	$1 + (0.0523 + 0.998i)T$
	11	$1 + (0.629 - 0.777i)T$
	13	$1 + (-0.920 - 0.390i)T$
	17	$1 + (-0.325 + 0.945i)T$
	23	$1 + (0.788 + 0.615i)T$
	29	$1 + (0.945 - 0.325i)T$
	31	$1 + (-0.913 - 0.406i)T$

Imaginary part of the first few zeros on the critical line

−18.61063548613437268112921464368, −17.7207874167065659718704269855, −16.80728156235331033288353843208, −16.12448970210050599926934637269, −15.331077097159691031144294379727, −14.69532592513821430920436915910, −14.04352447972198459520941898411, −13.8708991204595075695013258719, −12.78088604183849251183581616466, −12.438663428407319737942255917856, −11.532546475255333581423191035686, −10.652614320634832360495816083560, −10.10507681469183135377966377158, −9.402835202281598467577426993603, −8.79502989031210447838920066872, −7.43212736150117224173004609622, −7.12288122999548313047643770630, −6.56740383219420743083090910829, −5.0521847561420326669286103787, −4.68384381463828439436482484212, −3.936644849293648097253829909317, −3.280143932381177800398202643936, −2.377169344144298311040267817577, −1.78566500385530571241369933778, −0.7750131774257239363442594989, 1.416693684396338244402503713711, 2.28016354691296330700648087395, 2.98111033520976406606545634165, 3.54152281801780208766535670539, 4.466260168603644060988359761182, 5.17834767529327241224462580105, 6.07928927280562985984574576543, 6.63268627466857726429382356095, 7.59154586075674766329608419814, 8.14994016261974197053864472099, 8.84662508720975813810269109621, 9.40434585336698917361559129044, 10.48693680997702537131150488156, 11.40701645956781104838765378778, 12.08937963216464415841463184282, 12.78997833023173180553943679841, 13.2347451869044853641578170466, 14.171395359311876821504303832233, 14.58011341404048752677639127609, 15.26517886632088476515964342201, 15.62579161614779873829683697550, 16.51798256819533926030183577542, 17.30259883532125278273683235034, 17.93506353870740978298470163375, 19.06832705258542090627061975832

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