↑

Properties

Label 1-1323-1323.338-r1-0-0
Degree $1$
Conductor $1323$
Sign $-0.400 - 0.916i$
Analytic cond. $142.176$
Root an. cond. $142.176$
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  + (0.797 − 0.603i)2-s + (0.270 − 0.962i)4-s + (0.0249 + 0.999i)5-s + (−0.365 − 0.930i)8-s + (0.623 + 0.781i)10-s + (0.797 − 0.603i)11-s + (0.542 − 0.840i)13-s + (−0.853 − 0.521i)16-s + (0.900 − 0.433i)17-s + 19-s + (0.969 + 0.246i)20-s + (0.270 − 0.962i)22-s + (−0.698 − 0.715i)23-s + (−0.998 + 0.0498i)25-s + (−0.0747 − 0.997i)26-s + ⋯
L(s)  = 1  + (0.797 − 0.603i)2-s + (0.270 − 0.962i)4-s + (0.0249 + 0.999i)5-s + (−0.365 − 0.930i)8-s + (0.623 + 0.781i)10-s + (0.797 − 0.603i)11-s + (0.542 − 0.840i)13-s + (−0.853 − 0.521i)16-s + (0.900 − 0.433i)17-s + 19-s + (0.969 + 0.246i)20-s + (0.270 − 0.962i)22-s + (−0.698 − 0.715i)23-s + (−0.998 + 0.0498i)25-s + (−0.0747 − 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.400 - 0.916i$
Analytic conductor: \(142.176\)
Root analytic conductor: \(142.176\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (338, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (1:\ ),\ -0.400 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.065134388 - 3.156183273i\)
\(L(\frac12)\) \(\approx\) \(2.065134388 - 3.156183273i\)
\(L(1)\) \(\approx\) \(1.620711253 - 0.7934717966i\)
\(L(1)\) \(\approx\) \(1.620711253 - 0.7934717966i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.797 - 0.603i)T \)
5 \( 1 + (0.0249 + 0.999i)T \)
11 \( 1 + (0.797 - 0.603i)T \)
13 \( 1 + (0.542 - 0.840i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.698 - 0.715i)T \)
29 \( 1 + (-0.698 + 0.715i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.0747 - 0.997i)T \)
41 \( 1 + (0.853 - 0.521i)T \)
43 \( 1 + (0.878 - 0.478i)T \)
47 \( 1 + (-0.921 - 0.388i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (0.318 + 0.947i)T \)
61 \( 1 + (0.270 + 0.962i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.0747 - 0.997i)T \)
73 \( 1 + (-0.733 + 0.680i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.542 - 0.840i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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

−20.96974497974161955017799500580, −20.51905826718980205901169388751, −19.68820890881249001923319936900, −18.69665739234303899823731025418, −17.50218871756650154192332324398, −17.13222260522595010761443331114, −16.21327149035492551373658208548, −15.836710183185882034589302704120, −14.73996381046130801992022318298, −14.13341295079029459225916027199, −13.30699693819135690029770385377, −12.67448938427740270780494418323, −11.712717850421162248455370340166, −11.496791204917094732940171290983, −9.706621934552825714765479777569, −9.29718651081507889007797235332, −8.10286710912341367810648151195, −7.66667460317207630536417841330, −6.47125500756143728258734189473, −5.840536565742688594025955717716, −4.93394555576189278445213805025, −4.1261837228252544403291938094, −3.52162413534775398600933210136, −2.05953298851734508165017343486, −1.16498092650020428177442151246, 0.56591056747287744993826674106, 1.55084904009208402912182786940, 2.784546185132734569123932306831, 3.35489040491187312805857249406, 4.07199304895319042309435255301, 5.44243994300993156853297456735, 5.91007099969847219971762893256, 6.85891829039618274632478974388, 7.66129682922495252652031369315, 8.940475511322669610219968320, 9.85129613234862582326262848326, 10.60812233515306907483367754378, 11.19095565626500798325127455787, 11.99763028001975848610972665588, 12.705652875890720438082088622154, 13.87321637347061843094837262155, 14.156076649203543174143929386635, 14.867303195609612220139331717570, 15.830915306690672412069668368221, 16.40843315235746206233849516147, 17.84822969835200346946594194634, 18.321943777452699483444439931736, 19.1412874595187342194587936857, 19.79182371500143027042187437579, 20.5941878639163612407932352844

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