L-function 1-21e2-441.110-r0-0-0

• Label: 1-21e2-441.110-r0-0-0
• Degree: $1$
• Conductor: $441$
• Sign: $0.944 - 0.328i$
• Analytic cond.: $2.04799$
• Root an. cond.: $2.04799$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.988 − 0.149i)2^-s + (0.955 − 0.294i)4^-s + (−0.900 + 0.433i)5^-s + (0.900 − 0.433i)8^-s + (−0.826 + 0.563i)10^-s + (−0.623 − 0.781i)11^-s + (0.988 − 0.149i)13^-s + (0.826 − 0.563i)16^-s + (0.955 + 0.294i)17^-s + (0.5 + 0.866i)19^-s + (−0.733 + 0.680i)20^-s + (−0.733 − 0.680i)22^-s + (0.222 − 0.974i)23^-s + (0.623 − 0.781i)25^-s + (0.955 − 0.294i)26^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$441$    =    $3^{2} \cdot 7^{2}$
Sign:	$0.944 - 0.328i$
Analytic conductor:	$2.04799$
Root analytic conductor:	$2.04799$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{441} (110, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 441,\ (0:\ ),\ 0.944 - 0.328i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.262635699 - 0.3823645978i$
$L(1)$	$\approx$	$1.756795588 - 0.1759507745i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
good	2	$1 + (0.988 - 0.149i)T$
	5	$1 + (-0.900 + 0.433i)T$
	11	$1 + (-0.623 - 0.781i)T$
	13	$1 + (0.988 - 0.149i)T$
	17	$1 + (0.955 + 0.294i)T$
	19	$1 + (0.5 + 0.866i)T$
	23	$1 + (0.222 - 0.974i)T$
	29	$1 + (0.733 - 0.680i)T$
	31	$1 + (0.5 + 0.866i)T$

Imaginary part of the first few zeros on the critical line

−23.80893516646139473110209348792, −23.29431578072601095956191157859, −22.70594879387624971557101151357, −21.46791758679654584102891447373, −20.767500141415001818137997101521, −20.06507686343503978800316188252, −19.20181030745650105986579573507, −18.02991044858508037645118232788, −16.88773773958883066974912779856, −15.870121799750203377516333742615, −15.57935975950191033844151135299, −14.52343646980272350276317543742, −13.463073598142106089568594452385, −12.764306437283444938566019935103, −11.81455591313837038812182723157, −11.21390695284359374970992661586, −10.0070333894355593354901452414, −8.60484763186056841283688342928, −7.62957474628346761179848744059, −6.9429202339520937020812681103, −5.55638877723957010850035041308, −4.8022560058328119916363138240, −3.791837086192760785940157071060, −2.88667103928103259646759365258, −1.348452535056967044914511827273, 1.18832358395202863558477537732, 2.906129524858665102451192110196, 3.46730062966293712046846872257, 4.54418270163407543031672527784, 5.719542254817866731961433351526, 6.53170675378107750512742918784, 7.728352199192891644659950300644, 8.41393642384188500579291024676, 10.28157764315235791614279827086, 10.783293676698212433227461740840, 11.82191916360826925834922385143, 12.453346970358285641885492517954, 13.61512598461922315809791189702, 14.29191519416521468918438832637, 15.27475644474982768976247072259, 16.00238592570310838638332255066, 16.633774111980960374176870420703, 18.31669356647707114104461566522, 18.97624627918390162500474747663, 19.78884034508949704610725731912, 20.9136904369319711897079828110, 21.27307518877441183939242336589, 22.69675946036974746785193989383, 22.941695231132355824269335618148, 23.837663431911150476198657990323

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