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Properties

Label 1-837-837.689-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.930 - 0.365i$
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  + (−0.559 + 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.766 + 0.642i)5-s + (0.0348 + 0.999i)7-s + (0.978 + 0.207i)8-s + (−0.104 − 0.994i)10-s + (−0.0348 − 0.999i)11-s + (0.438 − 0.898i)13-s + (−0.848 − 0.529i)14-s + (−0.719 + 0.694i)16-s + (−0.309 − 0.951i)17-s + (0.913 + 0.406i)19-s + (0.882 + 0.469i)20-s + (0.848 + 0.529i)22-s + (−0.848 − 0.529i)23-s + ⋯
L(s)  = 1  + (−0.559 + 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.766 + 0.642i)5-s + (0.0348 + 0.999i)7-s + (0.978 + 0.207i)8-s + (−0.104 − 0.994i)10-s + (−0.0348 − 0.999i)11-s + (0.438 − 0.898i)13-s + (−0.848 − 0.529i)14-s + (−0.719 + 0.694i)16-s + (−0.309 − 0.951i)17-s + (0.913 + 0.406i)19-s + (0.882 + 0.469i)20-s + (0.848 + 0.529i)22-s + (−0.848 − 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.930 - 0.365i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.930 - 0.365i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07013863987 + 0.3701251721i\)
\(L(\frac12)\) \(\approx\) \(-0.07013863987 + 0.3701251721i\)
\(L(1)\) \(\approx\) \(0.5657560469 + 0.2797466823i\)
\(L(1)\) \(\approx\) \(0.5657560469 + 0.2797466823i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.559 + 0.829i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.0348 + 0.999i)T \)
11 \( 1 + (-0.0348 - 0.999i)T \)
13 \( 1 + (0.438 - 0.898i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.848 - 0.529i)T \)
29 \( 1 + (-0.559 + 0.829i)T \)
37 \( 1 + T \)
41 \( 1 + (0.719 + 0.694i)T \)
43 \( 1 + (-0.997 - 0.0697i)T \)
47 \( 1 + (0.241 + 0.970i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (-0.559 - 0.829i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.990 - 0.139i)T \)
83 \( 1 + (0.997 + 0.0697i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (-0.882 - 0.469i)T \)
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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

−21.2177578719789432500375557088, −20.46734015559011130904061396763, −19.92014255756429335667474980025, −19.399217994722432347281177698426, −18.356728606992130737484075259095, −17.515663494738106080881127857255, −16.827318493620320370696255007957, −16.13111631361798600661128320042, −15.188068391110506387105762294097, −13.88996939589422709794341003906, −13.21036788177746815140202242912, −12.37663645030158265892895991369, −11.57252742469224922261382348258, −10.95848586566814966525800102630, −9.863194592301278046463739748894, −9.28727523270362766596224107653, −8.15679072869083319637794026732, −7.60523174934782735767959673193, −6.69771506253312742806221977092, −4.99168423003979165781716959816, −4.04952104854478921552241815425, −3.72657904166186010624262103001, −2.08580763361733451636082709894, −1.2253689418378453282851896479, −0.13039557683648222704618696153, 0.97243502327955702240547902705, 2.571579589083543729472180393761, 3.50413569561296858433781272766, 4.85791757160763869750022297039, 5.79456945014751938871065787567, 6.424547120001185274300939818681, 7.61450788814625423637176269402, 8.131709645616188558722750195989, 8.97907583552667895438012634159, 9.899142438494051571093009760509, 10.9692468535505276628631900664, 11.49456253521220492183154480263, 12.65175573008205009279182071780, 13.81887192062352237284936744237, 14.5191370680954502947355504889, 15.32801872347212781674842768115, 16.018479822583171513029559913264, 16.42254896614493736748297766109, 17.94978966299068695779882963233, 18.30148517588256703923539207725, 18.86575410967133551589893808678, 19.81267487123367517378531798657, 20.54423271526696305424782458047, 22.05821229376155876352612392636, 22.357506034674997284234697432419

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