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Properties

Label 1-837-837.140-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.827 - 0.561i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (0.882 + 0.469i)2-s + (0.559 + 0.829i)4-s + (−0.766 + 0.642i)5-s + (0.961 + 0.275i)7-s + (0.104 + 0.994i)8-s + (−0.978 + 0.207i)10-s + (0.241 + 0.970i)11-s + (−0.882 + 0.469i)13-s + (0.719 + 0.694i)14-s + (−0.374 + 0.927i)16-s + (−0.913 − 0.406i)17-s + (0.669 + 0.743i)19-s + (−0.961 − 0.275i)20-s + (−0.241 + 0.970i)22-s + (−0.961 + 0.275i)23-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)2-s + (0.559 + 0.829i)4-s + (−0.766 + 0.642i)5-s + (0.961 + 0.275i)7-s + (0.104 + 0.994i)8-s + (−0.978 + 0.207i)10-s + (0.241 + 0.970i)11-s + (−0.882 + 0.469i)13-s + (0.719 + 0.694i)14-s + (−0.374 + 0.927i)16-s + (−0.913 − 0.406i)17-s + (0.669 + 0.743i)19-s + (−0.961 − 0.275i)20-s + (−0.241 + 0.970i)22-s + (−0.961 + 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.827 - 0.561i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.827 - 0.561i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6343569395 + 2.063137352i\)
\(L(\frac12)\) \(\approx\) \(-0.6343569395 + 2.063137352i\)
\(L(1)\) \(\approx\) \(1.127078750 + 0.9895266725i\)
\(L(1)\) \(\approx\) \(1.127078750 + 0.9895266725i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.882 + 0.469i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.961 + 0.275i)T \)
11 \( 1 + (0.241 + 0.970i)T \)
13 \( 1 + (-0.882 + 0.469i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.961 + 0.275i)T \)
29 \( 1 + (0.882 + 0.469i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.990 - 0.139i)T \)
43 \( 1 + (0.848 - 0.529i)T \)
47 \( 1 + (0.615 + 0.788i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.848 - 0.529i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (0.882 + 0.469i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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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

−21.61067450935682297630497967441, −20.59716399575620057618347457496, −19.954427548112261079962874413475, −19.53093904673675276920304674710, −18.441316934596243178793240111737, −17.38780765671954199712350311346, −16.51998081248084453876744224529, −15.53917637126702116510356428644, −15.068151677123963119169843553760, −13.93902437190027900639268341013, −13.49081485911662169043605334992, −12.30784930601736033433480543067, −11.796291351781182628057657002688, −11.05719203591625820226236048194, −10.25357692720306275040676854988, −8.995128524986740737714473857789, −8.11214388717965846738657360408, −7.225503184521803981389649454722, −6.08356575926427733404516435534, −5.00685595431330153752082017770, −4.508210589584666234972126105370, −3.565907129009286032822261742145, −2.475306293347408623601441592027, −1.256400000218752949010039679226, −0.3320790832150776572598812578, 1.84946006334993014405704626301, 2.6464167436486903956963044817, 3.91642235982149220269396938029, 4.5412789572561058237590559651, 5.39425141610451219379648699107, 6.62839934055012180694832719838, 7.33641695402220987389464142630, 7.91661676608447619654387027813, 8.99616503016757483209190503354, 10.31564632882524581630285890986, 11.28542650034329456966332196320, 12.06207241728875690757840343203, 12.35031596285310914037664787644, 13.99080382682222855657542727166, 14.241409915762823012050332521476, 15.18928459371380714563057070595, 15.64016931831680360261431467705, 16.62974121584131631145247096138, 17.70063075549425855872418887900, 18.11893436526085024876850983536, 19.44227585716730990032803215602, 20.17339745774893710220751849384, 20.89275046325343147038444522777, 21.99888485066409595718961060162, 22.32320660115609468454986513603

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