L-function 1-3895-3895.1073-r0-0-0

• Label: 1-3895-3895.1073-r0-0-0
• Degree: $1$
• Conductor: $3895$
• Sign: $0.705 + 0.708i$
• 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

+ (−0.961 − 0.275i)2^-s + (−0.573 − 0.819i)3^-s + (0.848 + 0.529i)4^-s + (0.325 + 0.945i)6^-s + (−0.933 + 0.358i)7^-s + (−0.669 − 0.743i)8^-s + (−0.342 + 0.939i)9^-s + (−0.0523 + 0.998i)11^-s + (−0.0523 − 0.998i)12^-s + (−0.325 − 0.945i)13^-s + (0.996 − 0.0871i)14^-s + (0.438 + 0.898i)16^-s + (−0.681 + 0.731i)17^-s + (0.587 − 0.809i)18^-s + (0.829 + 0.559i)21^-s + (0.325 − 0.945i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3895$    =    $5 \cdot 19 \cdot 41$
Sign:	$0.705 + 0.708i$
Analytic conductor:	$18.0883$
Root analytic conductor:	$18.0883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3895} (1073, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3895,\ (0:\ ),\ 0.705 + 0.708i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.4068178497 + 0.1690877144i$
$L(1)$	$\approx$	$0.4732039271 - 0.08648532110i$

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	$1 + (-0.961 - 0.275i)T$
	3	$1 + (-0.573 - 0.819i)T$
	7	$1 + (-0.933 + 0.358i)T$
	11	$1 + (-0.0523 + 0.998i)T$
	13	$1 + (-0.325 - 0.945i)T$
	17	$1 + (-0.681 + 0.731i)T$
	23	$1 + (0.0697 + 0.997i)T$
	29	$1 + (0.731 - 0.681i)T$
	31	$1 + (0.978 - 0.207i)T$

Imaginary part of the first few zeros on the critical line

−18.43663880167621435552080975679, −17.64099916981500797754784344230, −16.9281381620510626925287149057, −16.285599781250496220461889838729, −16.16968560611698563560816652984, −15.384480168649891000601823555725, −14.480105272465656833416947173236, −13.908875741245106103579030966113, −12.85182426504097802810803343155, −11.89336410443432911063587034605, −11.37674813185224058199779677177, −10.70539217202085072829602317015, −10.03510589840161224597428170036, −9.48060964226385759356951652009, −8.84119492052677543211831973809, −8.18669587932255312232955093778, −6.882942091874738077680041892699, −6.620869833221840600782885877090, −5.94592119012504233385692462040, −4.965986850959393592867242931620, −4.25090338135425024141857766097, −3.1266445172003805358116308620, −2.603520709306823014715445546746, −1.13120349827320923007346123584, −0.29814973198551866171216879944, 0.70980653218892957048093247310, 1.73221625098387730446528001135, 2.452798743534402850483097747063, 3.10797449744120178114224860110, 4.27865530398809904389044740342, 5.36978652361945990375395482617, 6.28111789629743097748630313037, 6.58460398096611748674684052295, 7.61596479631503574001529637873, 7.908811826392933882566319904438, 8.914110052672253336957420959182, 9.69435613691382032224158735761, 10.27266665751127174488883030456, 10.94178855873095429294607244563, 11.86475960395995742874253535985, 12.27774944096573888772788650830, 12.98337324638963907953566976637, 13.37224467178558198480531830730, 14.72214626192108290728286181523, 15.65647557921263146229167983532, 15.79230742237147917691742275932, 17.013172824259172602090025176887, 17.38016662735797060512831525806, 17.841244228455820273412226518909, 18.62568115115008100204550491738

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