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Properties

Label 1-837-837.308-r0-0-0
Degree $1$
Conductor $837$
Sign $0.0236 - 0.999i$
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  + (0.882 − 0.469i)2-s + (0.559 − 0.829i)4-s + (−0.766 − 0.642i)5-s + (0.961 − 0.275i)7-s + (0.104 − 0.994i)8-s + (−0.978 − 0.207i)10-s + (−0.241 + 0.970i)11-s + (0.882 + 0.469i)13-s + (0.719 − 0.694i)14-s + (−0.374 − 0.927i)16-s + (0.913 − 0.406i)17-s + (0.669 − 0.743i)19-s + (−0.961 + 0.275i)20-s + (0.241 + 0.970i)22-s + (0.961 + 0.275i)23-s + ⋯
L(s)  = 1  + (0.882 − 0.469i)2-s + (0.559 − 0.829i)4-s + (−0.766 − 0.642i)5-s + (0.961 − 0.275i)7-s + (0.104 − 0.994i)8-s + (−0.978 − 0.207i)10-s + (−0.241 + 0.970i)11-s + (0.882 + 0.469i)13-s + (0.719 − 0.694i)14-s + (−0.374 − 0.927i)16-s + (0.913 − 0.406i)17-s + (0.669 − 0.743i)19-s + (−0.961 + 0.275i)20-s + (0.241 + 0.970i)22-s + (0.961 + 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0236 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0236 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.0236 - 0.999i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (308, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.0236 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.861198493 - 1.817737863i\)
\(L(\frac12)\) \(\approx\) \(1.861198493 - 1.817737863i\)
\(L(1)\) \(\approx\) \(1.602173679 - 0.8301348163i\)
\(L(1)\) \(\approx\) \(1.602173679 - 0.8301348163i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.882 - 0.469i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.961 - 0.275i)T \)
11 \( 1 + (-0.241 + 0.970i)T \)
13 \( 1 + (0.882 + 0.469i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.961 + 0.275i)T \)
29 \( 1 + (-0.882 + 0.469i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.990 + 0.139i)T \)
43 \( 1 + (-0.848 - 0.529i)T \)
47 \( 1 + (0.615 - 0.788i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.848 + 0.529i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (-0.438 + 0.898i)T \)
83 \( 1 + (-0.882 + 0.469i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (-0.241 + 0.970i)T \)
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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

−22.36155379870868807911076979583, −21.71504418463754304455308638580, −20.78302860264686544497929686983, −20.36743757783302085194135014702, −18.870324838934563538938202802665, −18.55225240629674751390339436723, −17.37086809963918960181241814754, −16.49759122349768117037164743883, −15.72958750231319975562767945444, −14.982844837129164973922661242097, −14.40624062882321179947324726548, −13.59436191369696661943417525279, −12.64709203175557808237219590370, −11.61365235147001183266658113290, −11.25327624872859328257167582991, −10.35929228087797467126446032463, −8.615176624161943520800597757784, −8.03190569152432778227281705387, −7.416101096664671732571803098475, −6.174205389348475301426711349344, −5.56324669619903961075350651810, −4.517554437895157771828764494800, −3.452392646389602326658169092598, −2.97102887837798533365255128181, −1.422666419295340502160149313155, 1.019894558702899949503148748144, 1.83015911252883909978029066926, 3.233264854075019571620529575559, 4.08803637729765179138680606038, 4.93496003450337260952542321873, 5.43702439790634796713558310958, 7.01604511196102762902333037212, 7.5321866411418032119631491477, 8.74339766726528274685729215661, 9.67825415804054809365049790909, 10.827864726824642312215636958135, 11.47875062659164923847521078341, 12.08031748547432648930429073849, 13.018278157279383560679826705696, 13.709731700900319865569554332908, 14.71508457335502093631288533737, 15.29264729953353544146877542656, 16.14536477604803060736806994993, 16.96878416965365548409414557933, 18.16718636519189675276565595710, 18.873526105064794984415266498762, 19.94125257378496887188586174142, 20.50601540433919431130579795085, 20.94889348867612479680633089542, 21.84819999702586971868444117501

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