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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(2.119163092 + 0.8680521251i\)
\(L(\frac12)\) \(\approx\) \(2.119163092 + 0.8680521251i\)
\(L(1)\) \(\approx\) \(1.562089351 + 0.3184510603i\)
\(L(1)\) \(\approx\) \(1.562089351 + 0.3184510603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.987 + 0.156i)T \)
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 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.453 - 0.891i)T \)
41 \( 1 + (0.951 + 0.309i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.156 - 0.987i)T \)
59 \( 1 + (0.453 + 0.891i)T \)
61 \( 1 + (-0.453 + 0.891i)T \)
67 \( 1 + (0.156 + 0.987i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.156 - 0.987i)T \)
89 \( 1 + (0.951 - 0.309i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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