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Properties

Label 1-837-837.203-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.942 + 0.334i$
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  + (−0.961 − 0.275i)2-s + (0.848 + 0.529i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (0.913 − 0.406i)10-s + (−0.615 − 0.788i)11-s + (0.719 + 0.694i)13-s + (0.374 + 0.927i)14-s + (0.438 + 0.898i)16-s + (0.309 − 0.951i)17-s + (−0.104 + 0.994i)19-s + (−0.990 + 0.139i)20-s + (0.374 + 0.927i)22-s + (−0.374 − 0.927i)23-s + ⋯
L(s)  = 1  + (−0.961 − 0.275i)2-s + (0.848 + 0.529i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (0.913 − 0.406i)10-s + (−0.615 − 0.788i)11-s + (0.719 + 0.694i)13-s + (0.374 + 0.927i)14-s + (0.438 + 0.898i)16-s + (0.309 − 0.951i)17-s + (−0.104 + 0.994i)19-s + (−0.990 + 0.139i)20-s + (0.374 + 0.927i)22-s + (−0.374 − 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.942 + 0.334i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.942 + 0.334i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0008071802793 + 0.004692156478i\)
\(L(\frac12)\) \(\approx\) \(0.0008071802793 + 0.004692156478i\)
\(L(1)\) \(\approx\) \(0.4722433177 - 0.04923770576i\)
\(L(1)\) \(\approx\) \(0.4722433177 - 0.04923770576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.961 - 0.275i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.615 - 0.788i)T \)
11 \( 1 + (-0.615 - 0.788i)T \)
13 \( 1 + (0.719 + 0.694i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.374 - 0.927i)T \)
29 \( 1 + (0.961 + 0.275i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.438 + 0.898i)T \)
43 \( 1 + (0.241 + 0.970i)T \)
47 \( 1 + (0.997 + 0.0697i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (-0.961 + 0.275i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.882 + 0.469i)T \)
83 \( 1 + (-0.241 - 0.970i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (0.990 - 0.139i)T \)
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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

−21.60467950086907130536938062116, −20.72374833680729024050408169162, −20.003173437873151084702013455132, −19.32480211995561534073194965745, −18.67842103397642836481595782201, −17.7000023933768435936002444050, −17.132189981364793155432549329759, −15.92272380772918597083152653616, −15.557071553303299052794594564775, −15.15387262195943931722570246781, −13.611777964018733213469584764577, −12.518325871824299648904436353647, −12.07977675270615297944002634455, −10.94884535032998560221969881541, −10.18098991055146543387096794902, −9.19903692058925193082144745247, −8.52439228543063839849419143059, −7.8160413338588844272744453404, −6.90174363895915608492893029059, −5.82609442050118940882912805255, −5.09254376246578946677200624960, −3.67081943605194907404014984434, −2.59479991707705225403505217929, −1.41376669844862711532710296302, −0.003275321916823492692409178247, 1.24813836437650896838099364169, 2.79875081980432135617503119304, 3.37525013617406415955472030555, 4.35788334320249518451513957930, 6.14182078679127501503444497489, 6.74965010721272544924213133003, 7.69514854388373346274114780890, 8.303555611005749963035501875068, 9.34975035362773597421094076099, 10.42312188135781216384906625021, 10.74596526197461688522036137985, 11.72168788944820461372847253893, 12.44709177402375568961330028483, 13.64446075655069556356304970812, 14.3959418014348716076394869152, 15.718604645799999266893650604005, 16.192884419615467097609680818967, 16.70288854753488141884376085228, 17.96889171834719645917075115188, 18.73622181219812609598215183102, 19.03363329731714605405522606641, 20.02206721941480614985738396325, 20.67233822311759314538936438902, 21.49628125640929482517385139434, 22.54621462005592083701926368214

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