L-function 1-3895-3895.1088-r0-0-0

• Label: 1-3895-3895.1088-r0-0-0
• Degree: $1$
• Conductor: $3895$
• Sign: $0.0295 - 0.999i$
• 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

+ (−0.997 + 0.0697i)2^-s + (−0.422 − 0.906i)3^-s + (0.990 − 0.139i)4^-s + (0.484 + 0.874i)6^-s + (−0.629 + 0.777i)7^-s + (−0.978 + 0.207i)8^-s + (−0.642 + 0.766i)9^-s + (0.544 − 0.838i)11^-s + (−0.544 − 0.838i)12^-s + (0.484 + 0.874i)13^-s + (0.573 − 0.819i)14^-s + (0.961 − 0.275i)16^-s + (−0.920 − 0.390i)17^-s + (0.587 − 0.809i)18^-s + (0.970 + 0.241i)21^-s + (−0.484 + 0.874i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0295 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3895$    =    $5 \cdot 19 \cdot 41$
Sign:	$0.0295 - 0.999i$
Analytic conductor:	$18.0883$
Root analytic conductor:	$18.0883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3895} (1088, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3895,\ (0:\ ),\ 0.0295 - 0.999i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3922559992 - 0.4040186015i$
$L(1)$	$\approx$	$0.5283162255 - 0.1171934443i$

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	$1 + (-0.997 + 0.0697i)T$
	3	$1 + (-0.422 - 0.906i)T$
	7	$1 + (-0.629 + 0.777i)T$
	11	$1 + (0.544 - 0.838i)T$
	13	$1 + (0.484 + 0.874i)T$
	17	$1 + (-0.920 - 0.390i)T$
	23	$1 + (0.694 + 0.719i)T$
	29	$1 + (-0.390 - 0.920i)T$
	31	$1 + (-0.669 - 0.743i)T$

Imaginary part of the first few zeros on the critical line

−18.497151902034643894437148437673, −17.92543866567878480648538697198, −17.20161809009387977856965276767, −16.83319008837735313425070557776, −16.124472518497094958823304814806, −15.47949516410185311669240341322, −14.95486622630169521123069121688, −14.16423270046438473556129496547, −12.79936877270791894954747821708, −12.61511401392705956190560090902, −11.42750047453667844924033146593, −10.885851367327879592696739436480, −10.39330860463580657905664484024, −9.74601372090925321933675053075, −9.09527197669564938491006629063, −8.49548954723126650875628249872, −7.48055232066963192261615078957, −6.68430679744148168872324620474, −6.289725203550455751547669082300, −5.22800401308809884420657846584, −4.329535938378941395451980932431, −3.52124073330916276712912679952, −2.91643048042105872668148803738, −1.68270555696624042882124328425, −0.72203510226805053767138014208, 0.34469303463842568309413387603, 1.35672962643315084477091995270, 2.13392213599387674051943142566, 2.8304715578076783191002896470, 3.80456603168184459109214795848, 5.19964191137609530354533577916, 6.01767394457269885446816321627, 6.42880817777283697965007277750, 7.02915461771087238044882261899, 7.91812451897001988883409344566, 8.6374415038500405852585243960, 9.212208297401422769629592015605, 9.79187835581750263974954857393, 11.16827666005870001033247562848, 11.31313281600151806153516662271, 11.84421020477733845566931448712, 12.90321271086852602884753357407, 13.33625912466956765068389378138, 14.29887918483244146448100407409, 15.1192926084715378616850868481, 15.94971535178388862575658584518, 16.49010246298044414270622681209, 17.0139304017644603060082153388, 17.820159857166059307129641864444, 18.48975505992554872886257410794

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