↑

Properties

Label 1-804-804.659-r0-0-0
Degree $1$
Conductor $804$
Sign $0.617 - 0.786i$
Analytic cond. $3.73376$
Root an. cond. $3.73376$
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  + (−0.841 + 0.540i)5-s + (−0.928 − 0.371i)7-s + (0.0475 + 0.998i)11-s + (−0.995 + 0.0950i)13-s + (−0.235 − 0.971i)17-s + (−0.928 + 0.371i)19-s + (0.981 + 0.189i)23-s + (0.415 − 0.909i)25-s + (0.5 − 0.866i)29-s + (0.995 + 0.0950i)31-s + (0.981 − 0.189i)35-s + (−0.5 − 0.866i)37-s + (−0.723 + 0.690i)41-s + (0.959 + 0.281i)43-s + (−0.327 − 0.945i)47-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)5-s + (−0.928 − 0.371i)7-s + (0.0475 + 0.998i)11-s + (−0.995 + 0.0950i)13-s + (−0.235 − 0.971i)17-s + (−0.928 + 0.371i)19-s + (0.981 + 0.189i)23-s + (0.415 − 0.909i)25-s + (0.5 − 0.866i)29-s + (0.995 + 0.0950i)31-s + (0.981 − 0.189i)35-s + (−0.5 − 0.866i)37-s + (−0.723 + 0.690i)41-s + (0.959 + 0.281i)43-s + (−0.327 − 0.945i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(3.73376\)
Root analytic conductor: \(3.73376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 804,\ (0:\ ),\ 0.617 - 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6305718253 - 0.3068808307i\)
\(L(\frac12)\) \(\approx\) \(0.6305718253 - 0.3068808307i\)
\(L(1)\) \(\approx\) \(0.7373359674 + 0.003888399708i\)
\(L(1)\) \(\approx\) \(0.7373359674 + 0.003888399708i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
67 \( 1 \)
good5 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 + (-0.928 - 0.371i)T \)
11 \( 1 + (0.0475 + 0.998i)T \)
13 \( 1 + (-0.995 + 0.0950i)T \)
17 \( 1 + (-0.235 - 0.971i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
23 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.995 + 0.0950i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.723 + 0.690i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.327 - 0.945i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.235 - 0.971i)T \)
73 \( 1 + (0.0475 - 0.998i)T \)
79 \( 1 + (-0.580 - 0.814i)T \)
83 \( 1 + (-0.888 - 0.458i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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

−22.32333322275760048086210247439, −21.65834976952345172115034124925, −20.78292058508127433007637203466, −19.597540719251769864553864419141, −19.37801747534205879228133866510, −18.698022764938318022069594410037, −17.19308615520695872555808710976, −16.8471324418692629101455453863, −15.77833764428278172684387427334, −15.33028158610989270464111918712, −14.33389323625167656852526658447, −13.11449609773173747374007811284, −12.63815606658866024279854668494, −11.82751785724327137511964736870, −10.875919641493699513474316246489, −9.99452850513281953925635335062, −8.736320557690502344978334985230, −8.538241982271147305244063581799, −7.21146974469493245431183920848, −6.41851537999575313574701614462, −5.35943404620374761461280173197, −4.393721423615497599334165399595, −3.40798355583946932011305188893, −2.54356424822216653254695207079, −0.90573132508101757927653608548, 0.41663972563268877630407568037, 2.26255344323367120788767968345, 3.093263338902042613699727438532, 4.16624758745795063282621870035, 4.88136264936618025921858021682, 6.39182079048097183525870545410, 7.06159683674483936426779341735, 7.65566657731432641910161904595, 8.89349247362895839781712412878, 9.88509600214360294981266089040, 10.43658146172695395138898771938, 11.597930540820403783632717428039, 12.25603245051710421494663131602, 13.064902384679251775053939592646, 14.09238776435697474822132149340, 15.012557208353761781253529715018, 15.53223081609115037658095024999, 16.48515567661615560568463826136, 17.26013009389368788848248852002, 18.17666529890143286391277430958, 19.28897984251125245022835400856, 19.46436933028341281594551392536, 20.40185017124154651879404472225, 21.3464010567212443366779045662, 22.45157483801156696675124023979

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