L-function 1-800-800.381-r0-0-0

• Label: 1-800-800.381-r0-0-0
• Degree: $1$
• Conductor: $800$
• Sign: $0.331 - 0.943i$
• Analytic cond.: $3.71518$
• Root an. cond.: $3.71518$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.891 − 0.453i)3^-s − i·7^-s + (0.587 − 0.809i)9^-s + (−0.156 + 0.987i)11^-s + (0.156 + 0.987i)13^-s + (−0.309 − 0.951i)17^-s + (0.453 − 0.891i)19^-s + (−0.453 − 0.891i)21^-s + (−0.587 − 0.809i)23^-s + (0.156 − 0.987i)27^-s + (0.891 − 0.453i)29^-s + (0.309 + 0.951i)31^-s + (0.309 + 0.951i)33^-s + (0.987 − 0.156i)37^-s + (0.587 + 0.809i)39^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$800$    =    $2^{5} \cdot 5^{2}$
Sign:	$0.331 - 0.943i$
Analytic conductor:	$3.71518$
Root analytic conductor:	$3.71518$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{800} (381, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 800,\ (0:\ ),\ 0.331 - 0.943i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.616088455 - 1.145313394i$
$L(1)$	$\approx$	$1.368080227 - 0.4345488752i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
good	3	$1 + (0.891 - 0.453i)T$
	7	$1 - iT$
	11	$1 + (-0.156 + 0.987i)T$
	13	$1 + (0.156 + 0.987i)T$
	17	$1 + (-0.309 - 0.951i)T$
	19	$1 + (0.453 - 0.891i)T$
	23	$1 + (-0.587 - 0.809i)T$
	29	$1 + (0.891 - 0.453i)T$
	31	$1 + (0.309 + 0.951i)T$

Imaginary part of the first few zeros on the critical line

−22.0598873932975061520962070062, −21.65591973916300422170532878130, −20.90382351921010917748897391381, −19.97961404103297597260086587663, −19.31855963869479510539371194151, −18.53378286628732259676348957705, −17.783233684202836364900063779097, −16.498382299316728231625340402046, −15.842934089481936534720936578273, −15.132990128092978934362597078028, −14.45258491679902635497024657940, −13.47216557585835495026428103731, −12.81155916021645605973099868732, −11.74521468286124607995793457234, −10.75405177240989401214631632869, −9.93501923530478267988999851898, −9.07330718043912234495735246006, −8.18348749253856973386536176592, −7.83463169704723502742958536887, −6.13049531855631125885234636757, −5.56964162152594940677871247778, −4.33078594333958220893938520236, −3.29894724819435308824927768291, −2.657220379667556747355065355337, −1.43948151915806662767936372594, 0.869579691472170297448605563014, 2.05422331831308722840356174250, 2.91672772218807014332432575185, 4.183103575743962503345323630, 4.68697100582264074506854765134, 6.450018064162685479556072431200, 7.0914844552354611743815399057, 7.71750183141089928981764243347, 8.84105451413060663947780352005, 9.56812493182015934429433771594, 10.38535752630334179685372742141, 11.53777404616730432656495928067, 12.40709010789690891764426513719, 13.31497437765597812143730314981, 14.00423168576180692322526246078, 14.51260462562445849874050151381, 15.68887068097503394786535833475, 16.28028672387717799044224131329, 17.63108249316599864378208419105, 17.95127542203227275108508208240, 19.11077534975288521414584738024, 19.729578948303058427598273198971, 20.51801260997641467859183460912, 20.93246239191831563486577883585, 22.13141823267987946573813339901

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