L-function 1-3895-3895.1114-r0-0-0

• Label: 1-3895-3895.1114-r0-0-0
• Degree: $1$
• Conductor: $3895$
• Sign: $0.179 - 0.983i$
• 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 + (0.965 − 0.258i)3^-s + (−0.669 + 0.743i)4^-s + (−0.629 − 0.777i)6^-s + (−0.987 − 0.156i)7^-s + (0.951 + 0.309i)8^-s + (0.866 − 0.5i)9^-s + (0.891 − 0.453i)11^-s + (−0.453 + 0.891i)12^-s + (0.777 − 0.629i)13^-s + (0.258 + 0.965i)14^-s + (−0.104 − 0.994i)16^-s + (0.998 + 0.0523i)17^-s + (−0.809 − 0.587i)18^-s + (−0.994 + 0.104i)21^-s + (−0.777 − 0.629i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3895$    =    $5 \cdot 19 \cdot 41$
Sign:	$0.179 - 0.983i$
Analytic conductor:	$18.0883$
Root analytic conductor:	$18.0883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3895} (1114, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3895,\ (0:\ ),\ 0.179 - 0.983i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.682340011 - 1.403115613i$
$L(1)$	$\approx$	$1.105104261 - 0.6181486091i$

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	$1 + (0.965 - 0.258i)T$
	7	$1 + (-0.987 - 0.156i)T$
	11	$1 + (0.891 - 0.453i)T$
	13	$1 + (0.777 - 0.629i)T$
	17	$1 + (0.998 + 0.0523i)T$
	23	$1 + (-0.104 + 0.994i)T$
	29	$1 + (0.998 - 0.0523i)T$
	31	$1 + (0.309 - 0.951i)T$

Imaginary part of the first few zeros on the critical line

−18.61574038463760168213526898895, −18.20507282323787822526145542794, −17.089281442647863551128411186914, −16.461679922434794340300567613474, −16.0050989084204416890041568045, −15.36270985668401749270752302482, −14.603726494860805945396344487967, −14.06621500211560343500455463029, −13.57922547567437759124577655117, −12.63182112652778764308962952588, −12.03425772639863760068966370229, −10.59928377254045718844806766494, −10.18412447101343857046167053772, −9.37447412848552531767946145923, −8.92003143726556564064269627868, −8.38044568383038488407780554707, −7.40364037597465768429588215202, −6.80166753977664782587767129245, −6.23149002676165726364767095484, −5.278674650417433258286143372759, −4.244035887525144156489076988494, −3.82105976016406883873827168815, −2.83569429324407155138999732433, −1.785345709158918627204052431537, −0.88360460716598996590924337071, 0.95176440642709606738581778317, 1.28519266054289325054174311693, 2.5902700280043887682581060818, 3.12718668639830320993656345489, 3.72085553596443665568499096163, 4.32184733312297266415851123544, 5.71243818437777302273456017337, 6.4888167626127451731182414626, 7.42396349801433861630821761297, 8.085835585941484941035002104324, 8.69918132026854015390373033450, 9.51732959869873101880412747750, 9.83673259245621815366330390645, 10.63409231322611910469893983629, 11.57584206511293567944301270760, 12.23656222963825862738816037177, 12.90356653741236077204790003001, 13.568855927291718720124220998681, 13.926492524327035626956939856926, 14.854946677160613253229548063884, 15.74543847346020937168144302322, 16.38776744386977486082174080311, 17.16851981821823598464731243697, 17.91431412207593180534297931052, 18.70208304400975257826486617954

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