Properties

Label 1-837-837.518-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.414 + 0.909i$
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  + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (−0.173 − 0.984i)5-s + (−0.374 + 0.927i)7-s + (−0.669 − 0.743i)8-s + (−0.809 − 0.587i)10-s + (−0.615 − 0.788i)11-s + (−0.961 + 0.275i)13-s + (0.374 + 0.927i)14-s + (−0.997 − 0.0697i)16-s + (0.669 + 0.743i)17-s + (−0.809 − 0.587i)19-s + (−0.990 + 0.139i)20-s + (−0.990 − 0.139i)22-s + (−0.615 + 0.788i)23-s + ⋯
L(s)  = 1  + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (−0.173 − 0.984i)5-s + (−0.374 + 0.927i)7-s + (−0.669 − 0.743i)8-s + (−0.809 − 0.587i)10-s + (−0.615 − 0.788i)11-s + (−0.961 + 0.275i)13-s + (0.374 + 0.927i)14-s + (−0.997 − 0.0697i)16-s + (0.669 + 0.743i)17-s + (−0.809 − 0.587i)19-s + (−0.990 + 0.139i)20-s + (−0.990 − 0.139i)22-s + (−0.615 + 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.414 + 0.909i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (518, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.414 + 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1844612002 - 0.2868466581i\)
\(L(\frac12)\) \(\approx\) \(-0.1844612002 - 0.2868466581i\)
\(L(1)\) \(\approx\) \(0.8003131629 - 0.5604864643i\)
\(L(1)\) \(\approx\) \(0.8003131629 - 0.5604864643i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.719 - 0.694i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.374 + 0.927i)T \)
11 \( 1 + (-0.615 - 0.788i)T \)
13 \( 1 + (-0.961 + 0.275i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.615 + 0.788i)T \)
29 \( 1 + (-0.719 + 0.694i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.559 - 0.829i)T \)
43 \( 1 + (0.719 - 0.694i)T \)
47 \( 1 + (-0.559 + 0.829i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.961 + 0.275i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (-0.0348 - 0.999i)T \)
83 \( 1 + (-0.241 - 0.970i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (-0.374 + 0.927i)T \)
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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

−22.71897912938678134197431877825, −22.30121183390030645011959655604, −21.17793868623792061641301143324, −20.47086607218686627673757938610, −19.58823852844030574571383243448, −18.509885890500396968347014312756, −17.80125477797036187731630482020, −16.86358262462291939769880601670, −16.28454746482052060340868643973, −15.14247224129703595964273959073, −14.75188900049215941600719084931, −13.93090012734721457019152411691, −13.08646517214799149029693252712, −12.29777045655276153302331372203, −11.40633579142415419671866972440, −10.259262918588510453733826321605, −9.77785820205761357937125616921, −8.0763009806636450883550827702, −7.538833313728472874963955721978, −6.84434395152275400155691059533, −6.0278116550438780458195528903, −4.846451833358019673509819068480, −4.063623524499006700315518386306, −3.07783575363464225706871414558, −2.26763946320220878727848681856, 0.10903054686447136830105761756, 1.627595154651743372333832318547, 2.54453383768048683824978898032, 3.579270820232735472217692852844, 4.550683570900558954841046923000, 5.54539718089424321564004189272, 5.89332582055397253934603081082, 7.369951110508027077228444609067, 8.59101726260723244594896349782, 9.23974399216592063600602939129, 10.132807615004484950838657915627, 11.14909040099693297853692753118, 12.02518821557318601940042388550, 12.62280290206439236705136051779, 13.181539101700764189084040821562, 14.208569812324008241997739755880, 15.12528536429296718914787909357, 15.80917688553002521639112227709, 16.61092357762936437196106162505, 17.664024398264048053531791447751, 18.85868707904434807717470430753, 19.27748233191270498941554553869, 20.02164541344466622867243302868, 20.997247980759781682131316398707, 21.65470425393983636970977191377

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