↑

Properties

Label 1-621-621.65-r0-0-0
Degree $1$
Conductor $621$
Sign $-0.908 - 0.417i$
Analytic cond. $2.88391$
Root an. cond. $2.88391$
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  + (0.857 + 0.513i)2-s + (0.472 + 0.881i)4-s + (−0.266 + 0.963i)5-s + (−0.997 − 0.0634i)7-s + (−0.0475 + 0.998i)8-s + (−0.723 + 0.690i)10-s + (−0.987 − 0.158i)11-s + (−0.444 + 0.895i)13-s + (−0.823 − 0.567i)14-s + (−0.553 + 0.832i)16-s + (−0.786 − 0.618i)17-s + (0.786 − 0.618i)19-s + (−0.975 + 0.220i)20-s + (−0.766 − 0.642i)22-s + (−0.857 − 0.513i)25-s + (−0.841 + 0.540i)26-s + ⋯
L(s)  = 1  + (0.857 + 0.513i)2-s + (0.472 + 0.881i)4-s + (−0.266 + 0.963i)5-s + (−0.997 − 0.0634i)7-s + (−0.0475 + 0.998i)8-s + (−0.723 + 0.690i)10-s + (−0.987 − 0.158i)11-s + (−0.444 + 0.895i)13-s + (−0.823 − 0.567i)14-s + (−0.553 + 0.832i)16-s + (−0.786 − 0.618i)17-s + (0.786 − 0.618i)19-s + (−0.975 + 0.220i)20-s + (−0.766 − 0.642i)22-s + (−0.857 − 0.513i)25-s + (−0.841 + 0.540i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $-0.908 - 0.417i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ -0.908 - 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1934160644 + 0.8833651151i\)
\(L(\frac12)\) \(\approx\) \(-0.1934160644 + 0.8833651151i\)
\(L(1)\) \(\approx\) \(0.8736678591 + 0.6920021753i\)
\(L(1)\) \(\approx\) \(0.8736678591 + 0.6920021753i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.857 + 0.513i)T \)
5 \( 1 + (-0.266 + 0.963i)T \)
7 \( 1 + (-0.997 - 0.0634i)T \)
11 \( 1 + (-0.987 - 0.158i)T \)
13 \( 1 + (-0.444 + 0.895i)T \)
17 \( 1 + (-0.786 - 0.618i)T \)
19 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (0.999 - 0.0317i)T \)
31 \( 1 + (-0.975 - 0.220i)T \)
37 \( 1 + (-0.580 - 0.814i)T \)
41 \( 1 + (-0.967 + 0.251i)T \)
43 \( 1 + (-0.296 + 0.954i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (0.553 + 0.832i)T \)
61 \( 1 + (0.605 + 0.795i)T \)
67 \( 1 + (-0.630 - 0.776i)T \)
71 \( 1 + (0.327 + 0.945i)T \)
73 \( 1 + (-0.786 + 0.618i)T \)
79 \( 1 + (-0.110 + 0.993i)T \)
83 \( 1 + (0.967 + 0.251i)T \)
89 \( 1 + (-0.888 - 0.458i)T \)
97 \( 1 + (-0.902 - 0.429i)T \)
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

−22.35515059847994262615304440376, −21.91440534908196855401594496313, −20.645758123500362314148849392846, −20.30827233963428606753881470360, −19.50282779926409597886162593144, −18.697886242507611846129431489290, −17.549177884368063742664089298117, −16.389140407014568272301950807898, −15.69575237351974637881149230553, −15.17455166606401584630135887961, −13.80783769975170766999051134055, −13.07295087856613872996784591166, −12.541894962381744270320171706471, −11.85825729975732643096012176526, −10.52206039408243257876214574444, −10.00026163350662936145787242534, −8.90627976217788546646694868303, −7.78015254982813498965544165127, −6.66340389693967427408147057353, −5.5308912229212717191602820145, −4.99307148866034918908878463991, −3.79659909621298494072440474546, −2.980610434536014594225141280473, −1.78596978101213423219838203362, −0.30635522832559496183933709287, 2.42993836821716471286005042192, 2.98125098291230547936292119574, 4.01937837042218206028785165790, 5.070128546633397242895487606150, 6.15366612986900764006007906419, 7.00628061502065827852701030550, 7.43607163461655004468015064148, 8.77723777758815385087836243364, 9.88948707061748217532636967928, 10.96577645011200312545625203533, 11.702604788755312322368879897033, 12.68469392695214812124359307956, 13.58925743909795821711806549723, 14.13221750168761645580584790659, 15.226231573266242148254188321137, 15.85590284731882304655158283044, 16.425081374689350595892330248406, 17.661104138409596925016628494276, 18.43912977397772103632840317514, 19.43577668593149728430794782799, 20.19997131444732371687972855743, 21.34491678390218286198253379611, 22.07142126089898103888978597094, 22.58505325437468241306276252686, 23.4549111804372916610903553838

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