L-function 1-920-920.107-r1-0-0

• Label: 1-920-920.107-r1-0-0
• Degree: $1$
• Conductor: $920$
• Sign: $0.0313 + 0.999i$
• Analytic cond.: $98.8677$
• Root an. cond.: $98.8677$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.755 + 0.654i)3^-s + (−0.909 + 0.415i)7^-s + (0.142 + 0.989i)9^-s + (0.959 − 0.281i)11^-s + (0.909 + 0.415i)13^-s + (−0.540 − 0.841i)17^-s + (0.841 + 0.540i)19^-s + (−0.959 − 0.281i)21^-s + (−0.540 + 0.841i)27^-s + (0.841 − 0.540i)29^-s + (0.654 + 0.755i)31^-s + (0.909 + 0.415i)33^-s + (0.989 − 0.142i)37^-s + (0.415 + 0.909i)39^-s + (−0.142 + 0.989i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$920$    =    $2^{3} \cdot 5 \cdot 23$
Sign:	$0.0313 + 0.999i$
Analytic conductor:	$98.8677$
Root analytic conductor:	$98.8677$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{920} (107, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 920,\ (1:\ ),\ 0.0313 + 0.999i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.093995356 + 2.029397297i$
$L(1)$	$\approx$	$1.357717667 + 0.5028637672i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	23	$1$
good	3	$1 + (0.755 + 0.654i)T$
	7	$1 + (-0.909 + 0.415i)T$
	11	$1 + (0.959 - 0.281i)T$
	13	$1 + (0.909 + 0.415i)T$
	17	$1 + (-0.540 - 0.841i)T$
	19	$1 + (0.841 + 0.540i)T$
	29	$1 + (0.841 - 0.540i)T$
	31	$1 + (0.654 + 0.755i)T$
	37	$1 + (0.989 - 0.142i)T$

Imaginary part of the first few zeros on the critical line

−21.45057851528419322288938845929, −20.3521668021012937862952648084, −19.89947737828667643374649788290, −19.3180773190309393734714956301, −18.39909798283560852173840080633, −17.65340627284148264323130088994, −16.81529974124969426153469392103, −15.75220745170378809404997610150, −15.13645285846107461955475737774, −14.09010716952321929106893202528, −13.48344746574719293896433416393, −12.807469897385590399870052766630, −12.00639152673566492076527259272, −10.96850965142987374202634593866, −9.88321899844453103303164236563, −9.1744466969908714382763370219, −8.38471894456292145323235299818, −7.4277189759584121905189682230, −6.555081311467504790502670830122, −6.063112486181543370886550524302, −4.38614691319051123095991494009, −3.55605258291755973994161162912, −2.78789005474389216791802265384, −1.50496391499767367395347888424, −0.658104630126861162464807099385, 1.047973836668059111356767274270, 2.379350848841834541652249881331, 3.29507224028135076756340807014, 3.96044637948241411632597462302, 5.02642719419800271336439686303, 6.16728835739279493956019091174, 6.91568379735068057944924254936, 8.17558072675852407384439835899, 8.92747920434152353864744314957, 9.5293289354001327012901531407, 10.281285863597164830506678749481, 11.420648097359849887698833390362, 12.093483296131584092015233985842, 13.45103495840199932018873505092, 13.745439329403526780815615608963, 14.73146284284273674793716154956, 15.62221148944790197278290452327, 16.19567497812788635025777869862, 16.78712289678490094396650275151, 18.16413241821439338040898620489, 18.80191706879992331639526760923, 19.75668650694500785700659659259, 20.09874581432712513638363638397, 21.2114816970778637519584100958, 21.7457521856781083012027316562

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