Properties

Label 1-837-837.448-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.999 + 0.0258i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.999 + 0.0258i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (448, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.999 + 0.0258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02045087685 - 1.582370956i\)
\(L(\frac12)\) \(\approx\) \(0.02045087685 - 1.582370956i\)
\(L(1)\) \(\approx\) \(0.8437048986 - 0.9378574721i\)
\(L(1)\) \(\approx\) \(0.8437048986 - 0.9378574721i\)

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 \( 1 + (0.848 + 0.529i)T \)
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

−22.719742590655361748080654875179, −21.9236081787262403872814368057, −21.08632532000776967057116043843, −20.43043940129539227387548662701, −19.12118639393803535612120931414, −18.20995733294969195349703074264, −17.72731949364746746133084145273, −16.96437312815900840654837778828, −16.00979742581739238720136223832, −15.10440351478228191043746359342, −14.46134199304779022021673680331, −13.98543949135599106330354596452, −13.02728835598080038121933563559, −12.046953196245101572002063912127, −11.07498239834867398060310887936, −10.30441704138594505068534844885, −9.11960687859456049268585821165, −8.26705562831727903458326743045, −7.24354605185120729602085930258, −6.759695664674323873985705610957, −5.91130943911973806191614476724, −4.49333364943434465270614287720, −4.24848114963014365555357159195, −2.84733156462143540698160020357, −1.74074537732240156542702267605, 0.61436245768815144238628357012, 1.68589700198047703207220266825, 2.59744509368706945398657534527, 3.84639239917714332770991650203, 4.66321045518552754564189491151, 5.56336863313057317368798490053, 6.1093148217869787525147622632, 7.937635986401522212858284952236, 8.7669765633072892147813045853, 9.1738105935373261631057001210, 10.54776998963924876242146447926, 11.16149133652662424565172690509, 11.92951616779854629025521051730, 12.84857665418528597206080571145, 13.444075509625395469984707434767, 14.17918880846640436615286727591, 15.36818954855356708397654320811, 15.800063812971809186201887340833, 17.37547418879637377705896174118, 17.63536915284451680903731928499, 18.75169317329161777114028484395, 19.54201205627830843046537462662, 20.25237397051010597574808353705, 21.15182212298466174791273658752, 21.44360878043392475121315968812

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