↑

Properties

Label 1-837-837.578-r1-0-0
Degree $1$
Conductor $837$
Sign $0.928 + 0.372i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.719 + 0.694i)2-s + (0.0348 + 0.999i)4-s + (0.939 − 0.342i)5-s + (0.990 + 0.139i)7-s + (−0.669 + 0.743i)8-s + (0.913 + 0.406i)10-s + (−0.990 − 0.139i)11-s + (−0.241 − 0.970i)13-s + (0.615 + 0.788i)14-s + (−0.997 + 0.0697i)16-s + (−0.309 − 0.951i)17-s + (−0.104 − 0.994i)19-s + (0.374 + 0.927i)20-s + (−0.615 − 0.788i)22-s + (0.615 + 0.788i)23-s + ⋯
L(s)  = 1  + (0.719 + 0.694i)2-s + (0.0348 + 0.999i)4-s + (0.939 − 0.342i)5-s + (0.990 + 0.139i)7-s + (−0.669 + 0.743i)8-s + (0.913 + 0.406i)10-s + (−0.990 − 0.139i)11-s + (−0.241 − 0.970i)13-s + (0.615 + 0.788i)14-s + (−0.997 + 0.0697i)16-s + (−0.309 − 0.951i)17-s + (−0.104 − 0.994i)19-s + (0.374 + 0.927i)20-s + (−0.615 − 0.788i)22-s + (0.615 + 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.928 + 0.372i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.928 + 0.372i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.050717470 + 0.7819668184i\)
\(L(\frac12)\) \(\approx\) \(4.050717470 + 0.7819668184i\)
\(L(1)\) \(\approx\) \(1.869816145 + 0.5418692289i\)
\(L(1)\) \(\approx\) \(1.869816145 + 0.5418692289i\)

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 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (0.990 + 0.139i)T \)
11 \( 1 + (-0.990 - 0.139i)T \)
13 \( 1 + (-0.241 - 0.970i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.615 + 0.788i)T \)
29 \( 1 + (0.719 + 0.694i)T \)
37 \( 1 + T \)
41 \( 1 + (0.997 + 0.0697i)T \)
43 \( 1 + (0.961 - 0.275i)T \)
47 \( 1 + (-0.559 - 0.829i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (0.719 - 0.694i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.848 + 0.529i)T \)
83 \( 1 + (-0.961 + 0.275i)T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (-0.374 - 0.927i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.57236292769347146347604807462, −21.16482238146639036327236700132, −20.7449316287701150665228899883, −19.569630746260746735200815791767, −18.71621675292243684460899751133, −18.08969660621405870294944788121, −17.260635219218161157692819553316, −16.217856005015032819054117440218, −14.95516360706803958887357855937, −14.54788045418681163749305325734, −13.778686862343367390851045726267, −13.00082654921125157607496303122, −12.20152896790101997148282335837, −11.117182681065694006709624466939, −10.58968796987882989906477200833, −9.83943682321313529070136626438, −8.82339779458825494879120676555, −7.67511289318062173695263214487, −6.46925092128784149736678479448, −5.76691171751765287690085304678, −4.78866702042965844762802511411, −4.1037151553293750705366466232, −2.636195479418919667016720310182, −2.07745036672115818979962242739, −1.0947710881581488594152589726, 0.74663037110693858287612700114, 2.33467871287032510175472378206, 2.9248861533025314673013362405, 4.534053555070906490858843055894, 5.2019790525549446818734349246, 5.64438862035610677402218353304, 6.90839447463352164087043984610, 7.726855363270368945631725170693, 8.56596842690065318231964781161, 9.40129665433006002663544451574, 10.657315622500173329837009407673, 11.435872450153697999331631839727, 12.556424311377670106056530083213, 13.23617401498839662017522947471, 13.83519290642244654778579790677, 14.71999010120714467444339938775, 15.48566512318052608830374395820, 16.22315152957758272065753532951, 17.291991577213202087488892443177, 17.80979826160368885873851995, 18.29804662983957813002061969674, 19.96424778905652770043636566215, 20.68440525102545778077843390772, 21.38436950316114154579324649714, 21.84756691626775446017761022600

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