L-function 1-3895-3895.107-r0-0-0

• Label: 1-3895-3895.107-r0-0-0
• Degree: $1$
• Conductor: $3895$
• Sign: $0.770 + 0.637i$
• 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

+ (−0.406 − 0.913i)2^-s + (0.866 + 0.5i)3^-s + (−0.669 + 0.743i)4^-s + (0.104 − 0.994i)6^-s + (0.587 + 0.809i)7^-s + (0.951 + 0.309i)8^-s + (0.5 + 0.866i)9^-s + (−0.309 + 0.951i)11^-s + (−0.951 + 0.309i)12^-s + (0.994 + 0.104i)13^-s + (0.5 − 0.866i)14^-s + (−0.104 − 0.994i)16^-s + (0.743 − 0.669i)17^-s + (0.587 − 0.809i)18^-s + (0.104 + 0.994i)21^-s + (0.994 − 0.104i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3895$    =    $5 \cdot 19 \cdot 41$
Sign:	$0.770 + 0.637i$
Analytic conductor:	$18.0883$
Root analytic conductor:	$18.0883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3895} (107, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3895,\ (0:\ ),\ 0.770 + 0.637i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.196678232 + 0.7903494199i$
$L(1)$	$\approx$	$1.329221941 + 0.05088815652i$

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	$1 + (-0.406 - 0.913i)T$
	3	$1 + (0.866 + 0.5i)T$
	7	$1 + (0.587 + 0.809i)T$
	11	$1 + (-0.309 + 0.951i)T$
	13	$1 + (0.994 + 0.104i)T$
	17	$1 + (0.743 - 0.669i)T$
	23	$1 + (0.994 + 0.104i)T$
	29	$1 + (-0.669 + 0.743i)T$
	31	$1 + (-0.309 + 0.951i)T$

Imaginary part of the first few zeros on the critical line

−18.49269496531670534858135384203, −17.873776751597592514586985459859, −16.90908508740586171584114022985, −16.6470097020563318840900821366, −15.60385672714085201016712527372, −15.02398132411779399699857153534, −14.43324885310501607262120431156, −13.6875835037430139294732908713, −13.362436421590331824383732761931, −12.65930146659187108450461106948, −11.2637586502932839178989010582, −10.85339306617145778740794356959, −9.88189195729155003091114471138, −9.239868697984271248806297239286, −8.32707297602970613208921963926, −8.04496051553826244324828110065, −7.46029601852284796484582679049, −6.535957615886746841635084315213, −5.97479919043584065509378513568, −5.08580465756610294070670607262, −3.97869300309018986749183448892, −3.58686827397949880634579148647, −2.34317504233231585841264677834, −1.21431658509024990205884473190, −0.82717624726170577276275992510, 1.17719801172008467248410103008, 1.91829173689265056686423135549, 2.61280059625694938742472181557, 3.35034653326084989267288259825, 4.067075674990767355274834200033, 5.00332948557412729620393059322, 5.39181917345253836552874535394, 7.02861935897646486786494748900, 7.62190276017775579547570066127, 8.49033042773383428372435726272, 8.8685584034843623994633575929, 9.58539705696647219123683474452, 10.186418710941105466391913491014, 11.050933662311375745671330549630, 11.48292685220177273915137772067, 12.58638337417039579418330440565, 12.87521979417388994773353850263, 13.87125410671351479971015990936, 14.4309674365474784404243409874, 15.13687031112522129685138378560, 15.90048244344167714796022545369, 16.52438108117525637196027682582, 17.48484667901913024913052383580, 18.20749808557008983009848015626, 18.67706607342258048687461331524

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