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Properties

Label 1-837-837.133-r0-0-0
Degree $1$
Conductor $837$
Sign $0.365 + 0.930i$
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  + (0.559 + 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

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

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.365 + 0.930i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.365 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.877119874 + 1.279030132i\)
\(L(\frac12)\) \(\approx\) \(1.877119874 + 1.279030132i\)
\(L(1)\) \(\approx\) \(1.427429310 + 0.7096623165i\)
\(L(1)\) \(\approx\) \(1.427429310 + 0.7096623165i\)

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

−21.85533995882128108244791750895, −21.16845131051164665674179543351, −20.56718528780309650281152589893, −19.829848670086957851245749945295, −18.8896687159772326126155063247, −17.98429861143761937961053197018, −17.55635164774390847118354049380, −16.278396721394377198938801775297, −15.161613520759655727341176262370, −14.83845358448100243804805960375, −13.5292972582384445164556029572, −13.04696793394552205426879966287, −12.2332146714788897958097798107, −11.66211407082207836063833312308, −10.33156890697936523626298167030, −9.84737154647891995953792836450, −8.96771081746461460127646268862, −8.12873694830760786627445459488, −6.56058614567768268118495915544, −5.51548189399036467010579192556, −5.26476004184246288832766092129, −4.034406178487721168525227163157, −2.917478825843107828889593863793, −1.990818467244386277902899410046, −1.15293156457099860725493822412, 1.10348144752342115301697070288, 2.81924632658960950936419942813, 3.44174726677211426172422938104, 4.6308766187152207665753828720, 5.459437011801174645136320861608, 6.59255688293163571471210294634, 6.89650302917012652717178398789, 7.9551836139888661691816415150, 9.018584431891344457075000337389, 9.80069668079407064823055818054, 11.019304712750276418382084811213, 11.579102437510036239569460689886, 12.9750257697227698392541402647, 13.72392003408012696935825731704, 14.05155143380032575122064838340, 14.80288842236854131171098479763, 16.08245508913423305839432437887, 16.52360656803639388315486865828, 17.3008022013549748423863799981, 18.24401747413336526947928562393, 18.75349106801397245407521425864, 20.10918405065451277425627293299, 20.9931474195001269319069119196, 21.64333700798894693817356302892, 22.33967522106355755228732644153

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