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Properties

Label 1-861-861.47-r1-0-0
Degree $1$
Conductor $861$
Sign $-0.556 + 0.831i$
Analytic cond. $92.5273$
Root an. cond. $92.5273$
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  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.207 + 0.978i)5-s + (−0.951 + 0.309i)8-s + (0.978 + 0.207i)10-s + (0.838 − 0.544i)11-s + (−0.156 + 0.987i)13-s + (−0.104 + 0.994i)16-s + (0.544 + 0.838i)17-s + (−0.777 + 0.629i)19-s + (0.587 − 0.809i)20-s + (−0.156 − 0.987i)22-s + (0.913 + 0.406i)23-s + (−0.913 + 0.406i)25-s + (0.838 + 0.544i)26-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.207 + 0.978i)5-s + (−0.951 + 0.309i)8-s + (0.978 + 0.207i)10-s + (0.838 − 0.544i)11-s + (−0.156 + 0.987i)13-s + (−0.104 + 0.994i)16-s + (0.544 + 0.838i)17-s + (−0.777 + 0.629i)19-s + (0.587 − 0.809i)20-s + (−0.156 − 0.987i)22-s + (0.913 + 0.406i)23-s + (−0.913 + 0.406i)25-s + (0.838 + 0.544i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.556 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.556 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.556 + 0.831i$
Analytic conductor: \(92.5273\)
Root analytic conductor: \(92.5273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 861,\ (1:\ ),\ -0.556 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2223372882 + 0.4162862163i\)
\(L(\frac12)\) \(\approx\) \(0.2223372882 + 0.4162862163i\)
\(L(1)\) \(\approx\) \(1.022758301 - 0.2583950395i\)
\(L(1)\) \(\approx\) \(1.022758301 - 0.2583950395i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.406 - 0.913i)T \)
5 \( 1 + (0.207 + 0.978i)T \)
11 \( 1 + (0.838 - 0.544i)T \)
13 \( 1 + (-0.156 + 0.987i)T \)
17 \( 1 + (0.544 + 0.838i)T \)
19 \( 1 + (-0.777 + 0.629i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (-0.453 - 0.891i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (-0.358 - 0.933i)T \)
53 \( 1 + (-0.998 - 0.0523i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (0.994 + 0.104i)T \)
67 \( 1 + (0.0523 - 0.998i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.629 - 0.777i)T \)
97 \( 1 + (-0.891 + 0.453i)T \)
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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

−21.74919524715027873554811395283, −20.83368751271457407213529188994, −20.22017673407115463419041170245, −19.20956496319863721006384002196, −18.00877144351948232401419416566, −17.44162591600330975415586206555, −16.70747323176887851233361410942, −16.08300157199802910769500431104, −15.0922446002299000381386980192, −14.50120593729836116180109210343, −13.510566076247314308607830907535, −12.67388017382594010976999611551, −12.33330231053644966190823426808, −11.084318946969748060380675379582, −9.70583857789072739776950850215, −9.084146157026023698317430243833, −8.31074419598857876957138793622, −7.329452593521326742035990733859, −6.5592053045203676983650976410, −5.38598879402638150012939272587, −4.933344913358640611628395974463, −3.942306558603026458637916287650, −2.84177114721299640917916979168, −1.28796299069688054194319259711, −0.08825077527112188971529143280, 1.509956831720838027503299207719, 2.19698875308972141395235188206, 3.54681935169323908593018841301, 3.84118262800849386867920984660, 5.24814269980680172522024100016, 6.18531326976982982230059395531, 6.86354644191409367594763036278, 8.2629648854385062899196836547, 9.25119280565738011257309907605, 9.98294433931650553028168405138, 10.88928916394894285712291069654, 11.44825334175773585362915601286, 12.285784143260427381815751236706, 13.270257556987900252453795701166, 14.11329760912372754422707773920, 14.62295205201287826891816677868, 15.33833922999836765252561772160, 16.820722500966362789612037627111, 17.36821357300213112168355500959, 18.67966628230354144748113631044, 18.94795317672666869191263885574, 19.580049318083672067418877995708, 20.73534871616367051010403819463, 21.53874127502937680063403612023, 21.887689127697394862446528809616

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