L-function 1-261-261.16-r0-0-0

• Label: 1-261-261.16-r0-0-0
• Degree: $1$
• Conductor: $261$
• Sign: $0.902 + 0.430i$
• Analytic cond.: $1.21207$
• Root an. cond.: $1.21207$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.365 − 0.930i)2^-s + (−0.733 − 0.680i)4^-s + (−0.988 + 0.149i)5^-s + (−0.733 + 0.680i)7^-s + (−0.900 + 0.433i)8^-s + (−0.222 + 0.974i)10^-s + (0.0747 + 0.997i)11^-s + (0.826 − 0.563i)13^-s + (0.365 + 0.930i)14^-s + (0.0747 + 0.997i)16^-s + 17^-s + (−0.222 + 0.974i)19^-s + (0.826 + 0.563i)20^-s + (0.955 + 0.294i)22^-s + (0.365 + 0.930i)23^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$261$    =    $3^{2} \cdot 29$
Sign:	$0.902 + 0.430i$
Analytic conductor:	$1.21207$
Root analytic conductor:	$1.21207$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{261} (16, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 261,\ (0:\ ),\ 0.902 + 0.430i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7866426968 + 0.1778141233i$
$L(1)$	$\approx$	$0.8490180800 - 0.1784709181i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	29	$1$
good	2	$1 + (0.365 - 0.930i)T$
	5	$1 + (-0.988 + 0.149i)T$
	7	$1 + (-0.733 + 0.680i)T$
	11	$1 + (0.0747 + 0.997i)T$
	13	$1 + (0.826 - 0.563i)T$
	17	$1 + T$
	19	$1 + (-0.222 + 0.974i)T$
	23	$1 + (0.365 + 0.930i)T$
	31	$1 + (-0.988 + 0.149i)T$

Imaginary part of the first few zeros on the critical line

−25.97223214335637921651158131803, −24.76039730685925414109395918628, −23.769638017232896102670000364617, −23.36984369067496304375941039104, −22.464953133872458837869267976988, −21.44280965118943218523960937523, −20.34823336071075373655262423846, −19.149042515427463114820476647139, −18.54982697046990069554748001288, −16.99184611569668031796279155685, −16.389971221086538588660024666425, −15.76963542237296634809674284526, −14.63516731994990848363320094945, −13.667711268068855659405937418118, −12.857624181017955161297858537233, −11.76625272102015666495563200437, −10.64135311485379860570955478672, −9.086374542394553027488430555036, −8.347153814289380570655924443067, −7.19823943056259491033651553462, −6.45917672172729445897667997352, −5.15362987660245594772289771968, −3.872993551659599718829178578319, −3.32216715748406745384409780557, −0.53368027758536180237246547480, 1.51074772767116845516571734482, 3.103283480113539990436992588476, 3.72131241633415215989113819547, 5.07702457312248974704261307926, 6.17796529597864894645548550930, 7.65328433830935016919121484045, 8.82331247536525765907290762864, 9.879582081523251789465355034900, 10.79362459139067595185265584620, 12.02642678144740769236120916407, 12.39500676628227901162219069838, 13.47158110260839784822101279758, 14.8630634492655146025845351506, 15.34709759645687267159308709402, 16.551228579287005649498308029622, 18.09580885765667928409258287355, 18.75512571287771241584483629121, 19.58222481354180802459094972499, 20.38556336094520597928432567375, 21.293176891214386433719333184529, 22.4595549447622690863269061595, 23.0322772593567328660821765658, 23.629410120038388338764231716528, 25.10899427909467978167844956705, 25.90921873380537147047179834770

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