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Properties

Label 1-837-837.347-r0-0-0
Degree $1$
Conductor $837$
Sign $0.975 - 0.218i$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.766 + 0.642i)11-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.939 + 0.342i)16-s + 17-s + (−0.5 − 0.866i)19-s + (0.939 − 0.342i)20-s + (−0.173 − 0.984i)22-s + (0.173 + 0.984i)23-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.173 − 0.984i)5-s + (0.766 + 0.642i)7-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.766 + 0.642i)11-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)14-s + (−0.939 + 0.342i)16-s + 17-s + (−0.5 − 0.866i)19-s + (0.939 − 0.342i)20-s + (−0.173 − 0.984i)22-s + (0.173 + 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.975 - 0.218i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.975 - 0.218i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.160677223 - 0.1284699248i\)
\(L(\frac12)\) \(\approx\) \(1.160677223 - 0.1284699248i\)
\(L(1)\) \(\approx\) \(0.8699576697 - 0.1795531636i\)
\(L(1)\) \(\approx\) \(0.8699576697 - 0.1795531636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (0.766 + 0.642i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 - T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−22.44075981630263899162232662153, −21.19457777593796184496083943484, −20.51536907788118879160338480229, −19.47495862303198678421441393956, −18.803867697471240314790363530343, −18.25111844945767087757248543513, −17.297068827417158801449417938, −16.727301908017408695227846838817, −15.78592729396271846565022404930, −14.92172078860189475621371338080, −14.15258045998462221469667673999, −13.85747503782389058827490368534, −12.158425965969027488098460569894, −11.16031666784022826121943194631, −10.62209737966083072559551610682, −9.96105713924202006806602589424, −8.61288798624694920060092753450, −8.10016272297694857900263623231, −7.19538511779775029447862357169, −6.36201389704474819924149015293, −5.6636480334677857787783430266, −4.28905700359505781554749581296, −3.328095017026296322024103654456, −1.88827251437973898331727145474, −0.83245575314522215737732663820, 1.22568692560700738352418648621, 1.676987068024512803813255680, 3.07115848365376148952490590055, 4.18623135321066243072110733089, 4.94561392380772792062407117613, 6.21367988785994163758012208991, 7.436047560434333589862164566294, 8.2387737867580457361159768796, 9.015855692626701157635043569690, 9.45574288833833047093085331911, 10.722903061729436857904672648370, 11.582640568960083029310353536475, 12.11308583853142578887307264109, 12.87411863932078466706824018195, 13.8559361124831650786364723592, 15.011107419183339394418986069523, 15.93559526226954702755214422040, 16.61700997831932242435406783817, 17.6125004655256574454789040903, 17.899404387247456762317657188935, 19.161081493778014847005214744846, 19.59132506686095673637521219953, 20.59701522010310352343426874705, 21.14420665941240053136623208364, 21.66976448818661899000455626823

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