L-function 1-800-800.509-r0-0-0

• Label: 1-800-800.509-r0-0-0
• Degree: $1$
• Conductor: $800$
• Sign: $0.712 + 0.701i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$800$    =    $2^{5} \cdot 5^{2}$
Sign:	$0.712 + 0.701i$
Analytic conductor:	$3.71518$
Root analytic conductor:	$3.71518$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{800} (509, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 800,\ (0:\ ),\ 0.712 + 0.701i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.119163092 + 0.8680521251i$
$L(1)$	$\approx$	$1.562089351 + 0.3184510603i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−21.99103839136145855374437946097, −21.29978960633761672262209355037, −20.37032609790245466628607625342, −19.83148129991745219098464823658, −19.16399522131802084690712913583, −18.29130918021614001341387518062, −17.294886071844434974511277788573, −16.55409760905204961757968263321, −15.553693051999878286338472540, −14.81170284322817845454157385122, −13.814674528468108622665433734719, −13.506162789725863794422787221291, −12.60737592381962906760755834584, −11.25348660024489454527011193062, −10.751993555750342300541777866924, −9.4226131021966552978619365892, −8.9401760856586843467956161884, −8.03546653265548338936117712104, −6.93638866964943725122762640772, −6.54565522165457538581392016799, −4.922089417261864148203771613136, −3.82280109008155136313476690355, −3.420796603063674553948822597960, −1.94947026780164996108120762939, −1.06558360197959711751571670147, 1.46072217093745349782745739513, 2.3540593868332666957587272502, 3.37577786029322271577030244633, 4.20687635135974532312087790452, 5.35382985591893150926021764654, 6.38948031919337207804829824139, 7.39648127188514586461580084930, 8.35536862065451331755457165714, 9.10935944208441786018240100665, 9.60088603973476639913106484476, 10.83368398533913068481068333808, 11.71146167746012959772693834878, 12.78493219430439054791209030051, 13.30933809330217696750089094507, 14.59435955461620080320983567005, 14.81910679319313132166178546899, 15.8231550530063445965271237070, 16.46653349724972177081723449828, 17.80217861197258468864969172935, 18.4469527352740828785289134760, 19.19434663805197943821639236972, 20.02695561170507485573663997093, 20.76403706884909997679361943892, 21.36219333973167566543324488829, 22.44687871112083667462474939781

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