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Properties

Label 1-1155-1155.839-r0-0-0
Degree $1$
Conductor $1155$
Sign $-0.822 - 0.568i$
Analytic cond. $5.36379$
Root an. cond. $5.36379$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)13-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + 23-s + (−0.809 − 0.587i)26-s + (0.809 + 0.587i)29-s + (−0.309 + 0.951i)31-s + 32-s − 34-s + (0.809 + 0.587i)37-s + (−0.309 − 0.951i)38-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)13-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + (0.809 − 0.587i)19-s + 23-s + (−0.809 − 0.587i)26-s + (0.809 + 0.587i)29-s + (−0.309 + 0.951i)31-s + 32-s − 34-s + (0.809 + 0.587i)37-s + (−0.309 − 0.951i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.822 - 0.568i$
Analytic conductor: \(5.36379\)
Root analytic conductor: \(5.36379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1155,\ (0:\ ),\ -0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4328941046 - 1.388247558i\)
\(L(\frac12)\) \(\approx\) \(0.4328941046 - 1.388247558i\)
\(L(1)\) \(\approx\) \(0.8780899991 - 0.7082482286i\)
\(L(1)\) \(\approx\) \(0.8780899991 - 0.7082482286i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−21.652257762331902773788811775648, −21.11767715883722973438133527430, −20.10638757301836593494516532592, −18.98742520793375236357307136982, −18.502098607388881145540401312643, −17.513877707774981098884461936110, −16.84955640500284437563911805954, −16.213548175464101080200097494488, −15.32527341434325077893091861782, −14.7246038089273510587913120725, −13.8157677116765251532963627020, −13.27706215962461231304132228375, −12.31054683836532512656946194669, −11.57021920964636721054857837883, −10.45274806232104374160958592583, −9.41657254577945620582051698247, −8.767975966974652951285922740184, −7.88012171679820886004055928603, −7.076646098065714397771040353736, −6.232456225333034843159300696279, −5.535797023685955392609924605314, −4.42961454291933547077582411403, −3.83036946360074150011092532771, −2.67018224405591415100588720333, −1.23965475688483012505834838844, 0.60434070131683436319920659491, 1.62216268749909118166434111119, 2.9972948202472248373486451191, 3.236751524609398235825008332281, 4.76762323758181228563100036536, 5.10442027063247690820987235853, 6.27332520836865953758485960823, 7.30092936075304272336670748331, 8.45598184416035276310818184159, 9.1569450634427212424446448908, 10.04052709794583709546448834935, 10.77892554582701884880709010394, 11.54604529053303031524748730959, 12.25725524430670738741163389606, 13.24064645726785406050451296370, 13.621537329342679579328591241949, 14.65592312291284290817435391838, 15.3719348778491823033796872142, 16.256069184127939261032228467337, 17.40890750914791751261670428703, 18.124769463567623538966663724914, 18.615145806859086219788581283928, 19.769929769541588245999801089738, 20.13995790412044786520052685611, 20.87826221570491606092138609309

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