L-function 1-4160-4160.1309-r0-0-0

• Label: 1-4160-4160.1309-r0-0-0
• Degree: $1$
• Conductor: $4160$
• Sign: $-0.156 + 0.987i$
• Analytic cond.: $19.3189$
• Root an. cond.: $19.3189$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.130 + 0.991i)3^-s + (−0.258 − 0.965i)7^-s + (−0.965 + 0.258i)9^-s + (0.793 + 0.608i)11^-s + (0.866 + 0.5i)17^-s + (0.608 + 0.793i)19^-s + (0.923 − 0.382i)21^-s + (0.258 − 0.965i)23^-s + (−0.382 − 0.923i)27^-s + (0.130 + 0.991i)29^-s − 31^-s + (−0.5 + 0.866i)33^-s + (−0.608 + 0.793i)37^-s + (−0.258 + 0.965i)41^-s + (0.130 − 0.991i)43^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$4160$    =    $2^{6} \cdot 5 \cdot 13$
Sign:	$-0.156 + 0.987i$
Analytic conductor:	$19.3189$
Root analytic conductor:	$19.3189$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4160} (1309, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 4160,\ (0:\ ),\ -0.156 + 0.987i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.088532784 + 1.274679964i$
$L(1)$	$\approx$	$1.042884870 + 0.3939303784i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	13	$1$
good	3	$1 + (0.130 + 0.991i)T$
	7	$1 + (-0.258 - 0.965i)T$
	11	$1 + (0.793 + 0.608i)T$
	17	$1 + (0.866 + 0.5i)T$
	19	$1 + (0.608 + 0.793i)T$
	23	$1 + (0.258 - 0.965i)T$
	29	$1 + (0.130 + 0.991i)T$
	31	$1 - T$
	37	$1 + (-0.608 + 0.793i)T$

Imaginary part of the first few zeros on the critical line

−18.37868264460628191952272538643, −17.598497292227095526521707604446, −17.04643976595992302385645347977, −16.158363936691508582472598784720, −15.54407092414557225822895293244, −14.66487633427143479626887628387, −14.09151803153940394828767591708, −13.46443187527037751621096555040, −12.74046009445486350796537594440, −12.02020459555599496197946452308, −11.605247452552408727247969310373, −10.94545088934014269655259513798, −9.61783807338565237719812531117, −9.182752940469619014259345269624, −8.562013932197875593147986293414, −7.64178056975743002036114365814, −7.144610097005850262129367946810, −6.20779644399144726618888186086, −5.72036815534438936506172431659, −5.03235503461998334681178974030, −3.65301561398630459478940717724, −3.094733002883801923496883483896, −2.28988721187889294714811253532, −1.441138468253540276666479773934, −0.53513083821139811994291270920, 0.970606566304107172212808765869, 1.893388123534508874285964598219, 3.14443781443854868563794364099, 3.65714389184963793200009395663, 4.272243415234016053548509238905, 5.07953364296445219714292177536, 5.81954533352078099117363828287, 6.78732924497496883441057433802, 7.37779318416025539306417148230, 8.359296218892889961213149015602, 8.93571888663188678423481605961, 9.94213122279570987523931833983, 10.13087946775405581945635075669, 10.83925164100991446606376018106, 11.74727697923062015535705195324, 12.34045946878411598084139291365, 13.24853864517920886461283610327, 14.055307590000145729400346980602, 14.64418662342616608321676209391, 14.986062370364604242468463403280, 16.11301475274415022502349734981, 16.58986214466037776908129123112, 16.94299457303627081579467550359, 17.76058465449756409619061307533, 18.63375049601856548002886961073

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