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Properties

Label 1-847-847.250-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.800 + 0.599i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (−0.345 − 0.938i)2-s + (0.104 + 0.994i)3-s + (−0.761 + 0.647i)4-s + (0.398 + 0.917i)5-s + (0.897 − 0.441i)6-s + (0.870 + 0.491i)8-s + (−0.978 + 0.207i)9-s + (0.723 − 0.690i)10-s + (−0.723 − 0.690i)12-s + (−0.0285 + 0.999i)13-s + (−0.870 + 0.491i)15-s + (0.161 − 0.986i)16-s + (0.483 + 0.875i)17-s + (0.532 + 0.846i)18-s + (−0.905 + 0.424i)19-s + (−0.897 − 0.441i)20-s + ⋯
L(s)  = 1  + (−0.345 − 0.938i)2-s + (0.104 + 0.994i)3-s + (−0.761 + 0.647i)4-s + (0.398 + 0.917i)5-s + (0.897 − 0.441i)6-s + (0.870 + 0.491i)8-s + (−0.978 + 0.207i)9-s + (0.723 − 0.690i)10-s + (−0.723 − 0.690i)12-s + (−0.0285 + 0.999i)13-s + (−0.870 + 0.491i)15-s + (0.161 − 0.986i)16-s + (0.483 + 0.875i)17-s + (0.532 + 0.846i)18-s + (−0.905 + 0.424i)19-s + (−0.897 − 0.441i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.800 + 0.599i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.800 + 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2402406323 + 0.7208126595i\)
\(L(\frac12)\) \(\approx\) \(0.2402406323 + 0.7208126595i\)
\(L(1)\) \(\approx\) \(0.7472404210 + 0.2395001354i\)
\(L(1)\) \(\approx\) \(0.7472404210 + 0.2395001354i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.345 - 0.938i)T \)
3 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (0.398 + 0.917i)T \)
13 \( 1 + (-0.0285 + 0.999i)T \)
17 \( 1 + (0.483 + 0.875i)T \)
19 \( 1 + (-0.905 + 0.424i)T \)
23 \( 1 + (-0.888 - 0.458i)T \)
29 \( 1 + (-0.974 + 0.226i)T \)
31 \( 1 + (0.432 + 0.901i)T \)
37 \( 1 + (0.380 - 0.924i)T \)
41 \( 1 + (0.696 + 0.717i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.532 - 0.846i)T \)
53 \( 1 + (0.161 + 0.986i)T \)
59 \( 1 + (-0.272 - 0.962i)T \)
61 \( 1 + (0.640 + 0.768i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (-0.921 - 0.389i)T \)
73 \( 1 + (0.548 - 0.836i)T \)
79 \( 1 + (0.217 - 0.976i)T \)
83 \( 1 + (-0.254 + 0.967i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (-0.993 + 0.113i)T \)
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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

−22.05905522496501046007165321712, −20.74299071456562715715637264055, −20.06005792916283070821389166561, −19.23353197093991268813997368876, −18.42147632707483618011068872713, −17.64604817072797305998363165368, −17.15992417279370330524524756609, −16.34383517347202232973367953893, −15.40861077553102424325230595061, −14.51778915127822812853758879654, −13.60205990301794342811960901561, −13.116636624289855339112208595679, −12.333645555888365699996342509623, −11.202844443726185401302680190773, −9.890363442463542958846526715490, −9.24719622522250225947260987480, −8.179195680976565408220286276363, −7.86002000203504969155331338485, −6.76008509074761872220137018462, −5.83453223400495749043997115380, −5.31239118404218510275831401700, −4.12246390323838023491062824332, −2.537184961846018741565709399339, −1.36086861597786152486352023280, −0.38620437569289022883498563408, 1.80001028484890128150849611104, 2.559928385650666804859482767565, 3.700264701513302925885168852393, 4.15207374999370085370533965807, 5.43457285540267696792073481734, 6.447404846091960173960390460, 7.74873924562842846426043696497, 8.70204402782829529860878357962, 9.46502758278510993412848601649, 10.32011780953514611701955524329, 10.70065254247958064808908470320, 11.58213190268684530314012719744, 12.466089176931091027383863329949, 13.65069275761155465953016119420, 14.37445379490890536547147416219, 14.93193981972031562004769084000, 16.24175006943254637205973283718, 16.91605024699228489984180127103, 17.67686697136657372055010985740, 18.65512611244168303857271996804, 19.24209394248936390534625627838, 20.08956809010250766212304286806, 21.01700616809675274046314046035, 21.62486831113514964337158420330, 21.97270963229107563681305374349

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