L-function 1-10e2-100.11-r1-0-0

• Label: 1-10e2-100.11-r1-0-0
• Degree: $1$
• Conductor: $100$
• Sign: $-0.425 + 0.904i$
• Analytic cond.: $10.7464$
• Root an. cond.: $10.7464$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.809 + 0.587i)3^-s − 7^-s + (0.309 + 0.951i)9^-s + (−0.309 + 0.951i)11^-s + (0.309 + 0.951i)13^-s + (−0.809 + 0.587i)17^-s + (0.809 − 0.587i)19^-s + (−0.809 − 0.587i)21^-s + (−0.309 + 0.951i)23^-s + (−0.309 + 0.951i)27^-s + (−0.809 − 0.587i)29^-s + (0.809 − 0.587i)31^-s + (−0.809 + 0.587i)33^-s + (0.309 + 0.951i)37^-s + (−0.309 + 0.951i)39^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$100$    =    $2^{2} \cdot 5^{2}$
Sign:	$-0.425 + 0.904i$
Analytic conductor:	$10.7464$
Root analytic conductor:	$10.7464$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{100} (11, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 100,\ (1:\ ),\ -0.425 + 0.904i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.8766659312 + 1.381404465i$
$L(1)$	$\approx$	$1.081388548 + 0.5088628249i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
good	3	$1 + (0.809 + 0.587i)T$
	7	$1 - T$
	11	$1 + (-0.309 + 0.951i)T$
	13	$1 + (0.309 + 0.951i)T$
	17	$1 + (-0.809 + 0.587i)T$
	19	$1 + (0.809 - 0.587i)T$
	23	$1 + (-0.309 + 0.951i)T$
	29	$1 + (-0.809 - 0.587i)T$
	31	$1 + (0.809 - 0.587i)T$

Imaginary part of the first few zeros on the critical line

−29.460700846449877005306814306095, −28.61410971977832494221238475951, −26.94789320567263350275019167036, −26.262641828281791479616076171637, −25.12645808754741957099875940935, −24.43765355289426877635143857338, −23.1418415768954438213430350305, −22.12063142293760181826736299106, −20.66149888086729297366626143673, −19.88385278796923605576659389532, −18.80243520614971783169961517363, −18.02695019581774911426945875832, −16.34985778180431588617065991601, −15.44460274203615558660128542124, −14.02133146096858480011795136366, −13.22079360501403320305683916530, −12.23153643508396595840924423055, −10.58174974812249823811777080684, −9.26916468857723266957295390019, −8.231898837148160993256276256272, −6.97226152605682535644106722173, −5.76120343778232217872878691512, −3.6278295194698934810972369339, −2.63975559294522670082553493352, −0.64137602175394656600372314701, 2.11339860590120723361623431921, 3.520180667568283749031222535, 4.70779111615043734242485773058, 6.507893844168187548348781212775, 7.828921442575390296806767069154, 9.30204804026100103237633552484, 9.868203450849767536774734686318, 11.37097444406704119480845023184, 12.980353061389130288440278164480, 13.78784386483765987010101211287, 15.22930769967971878173273303110, 15.84984227122171565308816471233, 17.10740479751194009774167731158, 18.63529763095557945643152890254, 19.65527082418544219971175256652, 20.4480744630058923183436069852, 21.65999650981566356339488933406, 22.50956893417377456953233178968, 23.81812575852650052871137352290, 25.12369461741461590002489448169, 26.08803405214710040190901032169, 26.53137720985088388777766558092, 28.08373105489283670368226275825, 28.74431315818797254959667785543, 30.24615714896790073330705470208

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