L-function 1-1309-1309.41-r1-0-0

• Label: 1-1309-1309.41-r1-0-0
• Degree: $1$
• Conductor: $1309$
• Sign: $0.999 + 0.00154i$
• Analytic cond.: $140.671$
• Root an. cond.: $140.671$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.891 − 0.453i)2^-s + (−0.852 − 0.522i)3^-s + (0.587 − 0.809i)4^-s + (0.649 + 0.760i)5^-s + (−0.996 − 0.0784i)6^-s + (0.156 − 0.987i)8^-s + (0.453 + 0.891i)9^-s + (0.923 + 0.382i)10^-s + (−0.923 + 0.382i)12^-s + (−0.951 − 0.309i)13^-s + (−0.156 − 0.987i)15^-s + (−0.309 − 0.951i)16^-s + (0.809 + 0.587i)18^-s + (0.987 + 0.156i)19^-s + (0.996 − 0.0784i)20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00154i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1309$    =    $7 \cdot 11 \cdot 17$
Sign:	$0.999 + 0.00154i$
Analytic conductor:	$140.671$
Root analytic conductor:	$140.671$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1309} (41, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1309,\ (1:\ ),\ 0.999 + 0.00154i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.887841664 + 0.002234898438i$
$L(1)$	$\approx$	$1.460250376 - 0.3871896877i$

Euler product

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

	$p$	$F_p(T)$
bad	7	$1$
	11	$1$
	17	$1$
good	2	$1 + (0.891 - 0.453i)T$
	3	$1 + (-0.852 - 0.522i)T$
	5	$1 + (0.649 + 0.760i)T$
	13	$1 + (-0.951 - 0.309i)T$
	19	$1 + (0.987 + 0.156i)T$
	23	$1 + (-0.382 + 0.923i)T$
	29	$1 + (0.972 + 0.233i)T$
	31	$1 + (-0.996 + 0.0784i)T$
	37	$1 + (-0.233 + 0.972i)T$

Imaginary part of the first few zeros on the critical line

−21.00006415149873258207381852255, −20.41476443027731305150547408340, −19.49421702910443281033167134142, −17.88966609681813229022999313885, −17.68138083398270493465561109948, −16.62644576615388328775608470992, −16.31756251860657953499557779680, −15.59996761873811892373648128212, −14.48882601569246697408164490647, −14.0622225877745029624703802619, −12.794407441838227995440759116648, −12.50165287558128333842507681901, −11.662457456829806840484166417945, −10.83407641197574149066329518453, −9.79752830696899823963524242508, −9.1516364847537526616842045089, −7.9890368140974542276136483650, −7.003658995531973700765800085078, −6.16576232324609259440222110496, −5.48033677431496918279342943790, −4.755528770198324544049972706078, −4.21989178943597706369579542322, −2.9698180748972625987419516868, −1.86172877638101876883791600281, −0.50714951989162204535692280700, 0.924003984680444950494707345732, 1.89449227115671508578339425567, 2.6882759882245077688882727477, 3.67483396383603906883294154148, 4.97896261967683512499105753376, 5.472796251799318741463955975488, 6.26964969922625811368822921366, 7.07218969657007705766987032009, 7.667498126954677570902335476503, 9.45721474316008406084457867359, 10.15252332159292411716843326450, 10.783113307554647244432001729635, 11.6293843369785840405920480137, 12.21317264812006526126562955469, 13.0418364531453547872520690977, 13.7984460476306370621417866500, 14.3498820862964172982054652409, 15.31297387918061892147696102986, 16.10054623570205977611409361493, 17.07226874678997769810257313935, 17.882298250607341792950698829013, 18.45066427116622450707772778848, 19.35020453198838120863915182151, 19.93024368196295048367174215717, 21.083943556352606891970219702364

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