↑

Properties

Label 1-847-847.670-r0-0-0
Degree $1$
Conductor $847$
Sign $0.931 - 0.364i$
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  + (−0.0475 + 0.998i)2-s + (0.5 − 0.866i)3-s + (−0.995 − 0.0950i)4-s + (0.786 − 0.618i)5-s + (0.841 + 0.540i)6-s + (0.142 − 0.989i)8-s + (−0.5 − 0.866i)9-s + (0.580 + 0.814i)10-s + (−0.580 + 0.814i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)15-s + (0.981 + 0.189i)16-s + (0.235 − 0.971i)17-s + (0.888 − 0.458i)18-s + (0.235 + 0.971i)19-s + (−0.841 + 0.540i)20-s + ⋯
L(s)  = 1  + (−0.0475 + 0.998i)2-s + (0.5 − 0.866i)3-s + (−0.995 − 0.0950i)4-s + (0.786 − 0.618i)5-s + (0.841 + 0.540i)6-s + (0.142 − 0.989i)8-s + (−0.5 − 0.866i)9-s + (0.580 + 0.814i)10-s + (−0.580 + 0.814i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)15-s + (0.981 + 0.189i)16-s + (0.235 − 0.971i)17-s + (0.888 − 0.458i)18-s + (0.235 + 0.971i)19-s + (−0.841 + 0.540i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.931 - 0.364i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (670, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.931 - 0.364i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.758039945 - 0.3320976172i\)
\(L(\frac12)\) \(\approx\) \(1.758039945 - 0.3320976172i\)
\(L(1)\) \(\approx\) \(1.296320185 + 0.02338297586i\)
\(L(1)\) \(\approx\) \(1.296320185 + 0.02338297586i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0475 + 0.998i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.786 - 0.618i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.235 - 0.971i)T \)
19 \( 1 + (0.235 + 0.971i)T \)
23 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (-0.580 - 0.814i)T \)
37 \( 1 + (-0.995 + 0.0950i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.888 + 0.458i)T \)
53 \( 1 + (0.981 - 0.189i)T \)
59 \( 1 + (-0.0475 - 0.998i)T \)
61 \( 1 + (-0.888 - 0.458i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (-0.327 - 0.945i)T \)
79 \( 1 + (0.786 - 0.618i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (-0.723 - 0.690i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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

−21.91750739087141861411861697651, −21.25511345579231279904151093640, −20.80915453332091278756022473575, −19.73149494716034318007640901592, −19.31276914022629609587411951398, −18.175130820544181436431447548372, −17.56293674761406591120058944250, −16.7538249159858653348604290602, −15.46757425866872457638255438387, −14.809091987152247461507379016745, −13.94570423296098320011279102145, −13.34795246820209604191940374907, −12.456535278679692284813052385660, −11.1272454109505191088471776630, −10.5950079721010511528129044875, −10.08920325652005082634581654725, −9.06616167269531125657310990937, −8.55640346742754744030414280464, −7.328923401163905154216066647435, −5.84870501515419460132431578864, −5.11440731703649546030962009364, −4.03868633334529899482656797174, −3.064343109694166327721984161158, −2.57498645468339591357650639655, −1.29560073068822158570243417516, 0.89282022292330986033722579378, 1.83356302831284148122717357435, 3.20533258866492748222550727538, 4.42033294496193433175531898884, 5.473286396352731027873000049191, 6.20539601925191046748013237144, 7.06091029198133862147521323818, 7.82456122258031816408903408905, 8.95798325997018671473947675927, 9.12425817696113572229432261299, 10.245928882762590078227753452997, 11.79643786850679246543210020586, 12.60848418714087366898226736498, 13.432754504659623742087995926698, 13.9929351084282634026227555437, 14.54833913074589340295063266350, 15.72213149645180047271339374703, 16.56822277789378722740344135218, 17.20522525410146868441449337650, 18.08943697491722846516781858660, 18.63507715252984497677661169908, 19.409292532496960600910399168975, 20.60788333671688604100201206006, 21.12455544293272648403792051035, 22.25410841995670896948412638718

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