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Properties

Label 1-837-837.637-r1-0-0
Degree $1$
Conductor $837$
Sign $0.556 - 0.830i$
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  + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.173 − 0.984i)5-s + (0.848 + 0.529i)7-s + (−0.978 + 0.207i)8-s + (−0.809 − 0.587i)10-s + (−0.0348 + 0.999i)11-s + (−0.559 + 0.829i)13-s + (0.848 − 0.529i)14-s + (−0.241 + 0.970i)16-s + (0.978 − 0.207i)17-s + (−0.809 − 0.587i)19-s + (−0.882 + 0.469i)20-s + (0.882 + 0.469i)22-s + (−0.0348 − 0.999i)23-s + ⋯
L(s)  = 1  + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.173 − 0.984i)5-s + (0.848 + 0.529i)7-s + (−0.978 + 0.207i)8-s + (−0.809 − 0.587i)10-s + (−0.0348 + 0.999i)11-s + (−0.559 + 0.829i)13-s + (0.848 − 0.529i)14-s + (−0.241 + 0.970i)16-s + (0.978 − 0.207i)17-s + (−0.809 − 0.587i)19-s + (−0.882 + 0.469i)20-s + (0.882 + 0.469i)22-s + (−0.0348 − 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.556 - 0.830i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.556 - 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.281038253 - 1.217666775i\)
\(L(\frac12)\) \(\approx\) \(2.281038253 - 1.217666775i\)
\(L(1)\) \(\approx\) \(1.206605829 - 0.6567652928i\)
\(L(1)\) \(\approx\) \(1.206605829 - 0.6567652928i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.438 - 0.898i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (0.848 + 0.529i)T \)
11 \( 1 + (-0.0348 + 0.999i)T \)
13 \( 1 + (-0.559 + 0.829i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.0348 - 0.999i)T \)
29 \( 1 + (-0.438 + 0.898i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.961 + 0.275i)T \)
43 \( 1 + (-0.438 + 0.898i)T \)
47 \( 1 + (0.961 - 0.275i)T \)
53 \( 1 + (0.978 - 0.207i)T \)
59 \( 1 + (0.559 - 0.829i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (0.615 - 0.788i)T \)
83 \( 1 + (0.997 - 0.0697i)T \)
89 \( 1 + (-0.669 + 0.743i)T \)
97 \( 1 + (0.848 + 0.529i)T \)
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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

−22.15830612037221458565937615041, −21.39394041653505608395852624343, −20.8962416784780861521272182465, −19.4880214066841709373627312788, −18.707092518844655146447151677884, −17.858283657932778362356746950526, −17.20841600736679167528369710196, −16.53367915474226160791644423654, −15.354028164604996843372340078359, −14.81207024577738343100921269615, −14.08795723770466873078789010576, −13.52170968253902700961473967840, −12.44008131326638913903651000463, −11.44565261915169477040567105104, −10.60635937587300928611341375833, −9.72374689379376143276568417628, −8.42351023258811512896347230414, −7.71352060425052481397130278760, −7.15823654137156225623366193876, −5.82673005413325374039529746588, −5.57049619097114258470208018995, −4.109229177653932429460668613229, −3.44776393445989645434225369980, −2.314456699458961282083853261724, −0.62212476319682703587874668064, 0.86574241919529924379955092650, 1.86842851964621144541004367250, 2.51646727299765886014405679010, 4.09683219854374070135307822513, 4.7824840421850283678399540806, 5.3013147038881962161654817342, 6.51331420274664011247991431649, 7.86912189292099786938825494212, 8.84929892461825111208046917675, 9.450682490004602814916355835817, 10.34325977571082505495991041889, 11.40832042606750254723802284268, 12.17360611081075895271941002688, 12.602911483589608476856641183969, 13.54610254032150382634209261633, 14.64727146690383165831668754295, 14.89734628766388106454431811743, 16.25283132754626299927684751139, 17.15630883210729371092715409757, 17.95126845861529782230890913915, 18.72440048286890562120863854430, 19.643692779336813652065712611318, 20.412084892355013973256919956283, 21.0182468761023731298748629960, 21.5865083617583826027208408873

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