↑

Properties

Label 1-847-847.391-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.957 + 0.289i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (0.870 + 0.491i)2-s + (−0.309 − 0.951i)3-s + (0.516 + 0.856i)4-s + (−0.941 − 0.336i)5-s + (0.198 − 0.980i)6-s + (0.0285 + 0.999i)8-s + (−0.809 + 0.587i)9-s + (−0.654 − 0.755i)10-s + (0.654 − 0.755i)12-s + (0.0855 − 0.996i)13-s + (−0.0285 + 0.999i)15-s + (−0.466 + 0.884i)16-s + (−0.998 − 0.0570i)17-s + (−0.993 + 0.113i)18-s + (−0.254 + 0.967i)19-s + (−0.198 − 0.980i)20-s + ⋯
L(s)  = 1  + (0.870 + 0.491i)2-s + (−0.309 − 0.951i)3-s + (0.516 + 0.856i)4-s + (−0.941 − 0.336i)5-s + (0.198 − 0.980i)6-s + (0.0285 + 0.999i)8-s + (−0.809 + 0.587i)9-s + (−0.654 − 0.755i)10-s + (0.654 − 0.755i)12-s + (0.0855 − 0.996i)13-s + (−0.0285 + 0.999i)15-s + (−0.466 + 0.884i)16-s + (−0.998 − 0.0570i)17-s + (−0.993 + 0.113i)18-s + (−0.254 + 0.967i)19-s + (−0.198 − 0.980i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03302063792 + 0.2234814857i\)
\(L(\frac12)\) \(\approx\) \(0.03302063792 + 0.2234814857i\)
\(L(1)\) \(\approx\) \(0.9554650422 + 0.06007164400i\)
\(L(1)\) \(\approx\) \(0.9554650422 + 0.06007164400i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.870 + 0.491i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.941 - 0.336i)T \)
13 \( 1 + (0.0855 - 0.996i)T \)
17 \( 1 + (-0.998 - 0.0570i)T \)
19 \( 1 + (-0.254 + 0.967i)T \)
23 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (-0.774 + 0.633i)T \)
31 \( 1 + (-0.974 - 0.226i)T \)
37 \( 1 + (-0.921 + 0.389i)T \)
41 \( 1 + (-0.736 + 0.676i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.993 - 0.113i)T \)
53 \( 1 + (-0.466 - 0.884i)T \)
59 \( 1 + (0.736 + 0.676i)T \)
61 \( 1 + (-0.870 + 0.491i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.362 - 0.931i)T \)
73 \( 1 + (-0.985 - 0.170i)T \)
79 \( 1 + (-0.610 + 0.791i)T \)
83 \( 1 + (0.696 - 0.717i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.941 + 0.336i)T \)
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

−21.97193492068366318327759887266, −21.07763975594718713996702784550, −20.277179127275740193943840817890, −19.58501851111087052090355841984, −18.90898293148737696071579814437, −17.72975682134545292027141266247, −16.66739214032938014497744769127, −15.63385115107353496103843719627, −15.51838651619384101747272444436, −14.529235872178475408811056373289, −13.77769804730018896386608433455, −12.68137854774622284019387151751, −11.690456541903226401062843537277, −11.23389315536570631838847957523, −10.66448331051827243935272483737, −9.50968195470761305570455174440, −8.80291027341687693723406603485, −7.27366030946793406195861630332, −6.52062342916023624481267018344, −5.44578799312680696564605412015, −4.50028505557830799853645049006, −3.94719673901922843381509614606, −3.13215699744467921500466210603, −1.94529772974658178131384180164, −0.07283971088527305270680609599, 1.63099507773758086002153106767, 2.83952880318940121587236439098, 3.80267620343647884140515854734, 4.84723152840582825182119400356, 5.66714156455165151115970220916, 6.603256990439348543641018411209, 7.37854023725225038062005356909, 8.1430880801935566202258152253, 8.72240031520519113807919193839, 10.626043181757357360436552692383, 11.343240179128857815278958206, 12.17848657053389702910025566430, 12.80366575218494750422849647117, 13.311401017261613369165120749109, 14.504063971318874286735328035078, 15.06890521518543572345988223899, 16.15296417262570440433645350801, 16.63134109898183790965262563119, 17.61754944947301747278434802811, 18.37493969597559392554148576821, 19.390642659060851450230310291298, 20.23532445044557299633560368850, 20.685400061570894772937076850271, 22.15726185373215335470763087150, 22.6428102403260114081320715788

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