Properties

Label 1-837-837.353-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.679 + 0.733i$
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  + (−0.0348 + 0.999i)2-s + (−0.997 − 0.0697i)4-s + (0.939 − 0.342i)5-s + (−0.719 − 0.694i)7-s + (0.104 − 0.994i)8-s + (0.309 + 0.951i)10-s + (−0.241 + 0.970i)11-s + (−0.848 + 0.529i)13-s + (0.719 − 0.694i)14-s + (0.990 + 0.139i)16-s + (−0.104 + 0.994i)17-s + (0.309 + 0.951i)19-s + (−0.961 + 0.275i)20-s + (−0.961 − 0.275i)22-s + (−0.241 − 0.970i)23-s + ⋯
L(s)  = 1  + (−0.0348 + 0.999i)2-s + (−0.997 − 0.0697i)4-s + (0.939 − 0.342i)5-s + (−0.719 − 0.694i)7-s + (0.104 − 0.994i)8-s + (0.309 + 0.951i)10-s + (−0.241 + 0.970i)11-s + (−0.848 + 0.529i)13-s + (0.719 − 0.694i)14-s + (0.990 + 0.139i)16-s + (−0.104 + 0.994i)17-s + (0.309 + 0.951i)19-s + (−0.961 + 0.275i)20-s + (−0.961 − 0.275i)22-s + (−0.241 − 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.679 + 0.733i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.679 + 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4163211214 + 0.9529296810i\)
\(L(\frac12)\) \(\approx\) \(0.4163211214 + 0.9529296810i\)
\(L(1)\) \(\approx\) \(0.7897809768 + 0.4845721879i\)
\(L(1)\) \(\approx\) \(0.7897809768 + 0.4845721879i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.0348 + 0.999i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.719 - 0.694i)T \)
11 \( 1 + (-0.241 + 0.970i)T \)
13 \( 1 + (-0.848 + 0.529i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.241 - 0.970i)T \)
29 \( 1 + (0.0348 - 0.999i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.374 - 0.927i)T \)
43 \( 1 + (-0.0348 + 0.999i)T \)
47 \( 1 + (0.374 + 0.927i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (-0.848 + 0.529i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (0.997 - 0.0697i)T \)
83 \( 1 + (-0.882 + 0.469i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (-0.719 - 0.694i)T \)
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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

−21.77713971771528408423179517380, −21.325993045815920991878446669330, −20.11548880275294377751415682278, −19.58189513208074394388768461588, −18.578017463940200778604329650819, −18.14671796749257825622417912314, −17.31814048233234812055151901216, −16.34382746879893041500933056581, −15.31224779730068342738284421267, −14.267446383814064002066432451, −13.54795501252880351463229040161, −12.944618586209738414263040461315, −12.03305521480742298430232395017, −11.15077016390951885183685589659, −10.36847863419155930818344504903, −9.334569574698574403815908435311, −9.21207069565556809877406011988, −7.83965281971325246424785757279, −6.64474137867284403362064086769, −5.48772697916983903166194577785, −5.08627060067622341541232314279, −3.37283777022285395278326719926, −2.8434995754990048055254359305, −2.018567535042187108289296975765, −0.50802470035214033775748509744, 1.23370250330999639224870129812, 2.5232119610730204972071943458, 4.0827454184279478148019357928, 4.65374460100293455425334374559, 5.84736238147980434303956719071, 6.44614942250412583767273564984, 7.34090896438878092855994461073, 8.19174510125148409588128833761, 9.36232047996588253620353482417, 9.87449771965351815908532957825, 10.48837400276303869587669394003, 12.37821973463979951850141838189, 12.75053638622106647644904762559, 13.72928936591206916703773537511, 14.34004656277551724816936875076, 15.17710699667989411692467344502, 16.18335983520199954294189803430, 16.97831665066218903671595756717, 17.24935691745606447573476981432, 18.25009108225534860454041547202, 19.077633053528107670620571822219, 20.031244285865901960400676523414, 20.916178737267121738642261518452, 21.865537778459016743031769594625, 22.57759882948082928326353540637

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