Properties

Label 1-837-837.142-r0-0-0
Degree $1$
Conductor $837$
Sign $0.979 + 0.201i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (−0.374 − 0.927i)2-s + (−0.719 + 0.694i)4-s + (0.173 − 0.984i)5-s + (−0.997 + 0.0697i)7-s + (0.913 + 0.406i)8-s + (−0.978 + 0.207i)10-s + (−0.997 + 0.0697i)11-s + (−0.615 − 0.788i)13-s + (0.438 + 0.898i)14-s + (0.0348 − 0.999i)16-s + (−0.809 + 0.587i)17-s + (0.669 + 0.743i)19-s + (0.559 + 0.829i)20-s + (0.438 + 0.898i)22-s + (0.438 + 0.898i)23-s + ⋯
L(s)  = 1  + (−0.374 − 0.927i)2-s + (−0.719 + 0.694i)4-s + (0.173 − 0.984i)5-s + (−0.997 + 0.0697i)7-s + (0.913 + 0.406i)8-s + (−0.978 + 0.207i)10-s + (−0.997 + 0.0697i)11-s + (−0.615 − 0.788i)13-s + (0.438 + 0.898i)14-s + (0.0348 − 0.999i)16-s + (−0.809 + 0.587i)17-s + (0.669 + 0.743i)19-s + (0.559 + 0.829i)20-s + (0.438 + 0.898i)22-s + (0.438 + 0.898i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.979 + 0.201i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.979 + 0.201i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4867921999 + 0.04950929875i\)
\(L(\frac12)\) \(\approx\) \(0.4867921999 + 0.04950929875i\)
\(L(1)\) \(\approx\) \(0.5657858152 - 0.2613488646i\)
\(L(1)\) \(\approx\) \(0.5657858152 - 0.2613488646i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.374 + 0.927i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (0.997 - 0.0697i)T \)
11 \( 1 + (0.997 - 0.0697i)T \)
13 \( 1 + (0.615 + 0.788i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (-0.438 - 0.898i)T \)
29 \( 1 + (0.374 + 0.927i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.0348 - 0.999i)T \)
43 \( 1 + (-0.990 - 0.139i)T \)
47 \( 1 + (0.882 - 0.469i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (0.374 - 0.927i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.961 + 0.275i)T \)
83 \( 1 + (-0.990 - 0.139i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (-0.559 - 0.829i)T \)
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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

−22.27359169867937854366041971239, −21.72043749022304017548953623414, −20.30060769541961072994298616695, −19.44090670002988069819984622054, −18.68089430804216752056739294, −18.21668383856297474664492082354, −17.309598321311109648673342005651, −16.359227310034531681778093939968, −15.75982416240521019260047682418, −15.016987239007744470151869687297, −14.0984666572308180009355310693, −13.4815186826394591671109673375, −12.60484991408814047342740721549, −11.15727707530390842163844738511, −10.486560836910457807493035549024, −9.556366126064012084227509056846, −9.03826850988735372759522181155, −7.67351256139812583085693944249, −6.99100296343936053736859445846, −6.472173668366113680665567847930, −5.40884462474768331043528748504, −4.46226724937107757206475365822, −3.13584151297521990439739938097, −2.23130826795696511188939671480, −0.30488666444428345369955994984, 0.960699829137429616230933564147, 2.2231898072340329046880854955, 3.09185107196353009258616806424, 4.139591275945099344827745133971, 5.12843066565440599717950999941, 5.97381189110824488718923762343, 7.56682690004219897326972730784, 8.131870413526842762305277249786, 9.26197448358737635784624228769, 9.759015790546658036867109747521, 10.52422420475400397223032689848, 11.62557729775477230737769628015, 12.49608681443844969607122366411, 13.13624654093151040628626111750, 13.450210947153614146092299950710, 14.99552466027979908864319673073, 15.99352856611274166181093120296, 16.63632104738775580607348307742, 17.55932308541408506950527084260, 18.13469846172696036080931663320, 19.30982389964888773121913550260, 19.69904059890432382533544990292, 20.5687283116770702876157041990, 21.151503719769346833460992861568, 22.050360763556412260136072010028

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