↑

Properties

Label 1-3381-3381.1076-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.900 + 0.435i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (−0.390 − 0.920i)2-s + (−0.694 + 0.719i)4-s + (−0.999 + 0.0135i)5-s + (0.933 + 0.359i)8-s + (0.403 + 0.915i)10-s + (0.476 + 0.879i)11-s + (−0.917 − 0.396i)13-s + (−0.0339 − 0.999i)16-s + (−0.275 + 0.961i)17-s + (−0.580 + 0.814i)19-s + (0.685 − 0.728i)20-s + (0.623 − 0.781i)22-s + (0.999 − 0.0271i)25-s + (−0.00679 + 0.999i)26-s + (−0.970 + 0.242i)29-s + ⋯
L(s)  = 1  + (−0.390 − 0.920i)2-s + (−0.694 + 0.719i)4-s + (−0.999 + 0.0135i)5-s + (0.933 + 0.359i)8-s + (0.403 + 0.915i)10-s + (0.476 + 0.879i)11-s + (−0.917 − 0.396i)13-s + (−0.0339 − 0.999i)16-s + (−0.275 + 0.961i)17-s + (−0.580 + 0.814i)19-s + (0.685 − 0.728i)20-s + (0.623 − 0.781i)22-s + (0.999 − 0.0271i)25-s + (−0.00679 + 0.999i)26-s + (−0.970 + 0.242i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.900 + 0.435i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1076, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.900 + 0.435i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03485793443 + 0.1519566933i\)
\(L(\frac12)\) \(\approx\) \(0.03485793443 + 0.1519566933i\)
\(L(1)\) \(\approx\) \(0.5659752351 - 0.09970074898i\)
\(L(1)\) \(\approx\) \(0.5659752351 - 0.09970074898i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.390 - 0.920i)T \)
5 \( 1 + (-0.999 + 0.0135i)T \)
11 \( 1 + (0.476 + 0.879i)T \)
13 \( 1 + (-0.917 - 0.396i)T \)
17 \( 1 + (-0.275 + 0.961i)T \)
19 \( 1 + (-0.580 + 0.814i)T \)
29 \( 1 + (-0.970 + 0.242i)T \)
31 \( 1 + (0.786 + 0.618i)T \)
37 \( 1 + (-0.0882 + 0.996i)T \)
41 \( 1 + (0.488 - 0.872i)T \)
43 \( 1 + (-0.933 + 0.359i)T \)
47 \( 1 + (0.826 + 0.563i)T \)
53 \( 1 + (0.942 + 0.333i)T \)
59 \( 1 + (-0.403 - 0.915i)T \)
61 \( 1 + (-0.869 - 0.494i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.947 + 0.320i)T \)
73 \( 1 + (-0.855 - 0.517i)T \)
79 \( 1 + (-0.327 + 0.945i)T \)
83 \( 1 + (0.986 + 0.162i)T \)
89 \( 1 + (0.976 - 0.215i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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

−18.56154379437558037983709813352, −17.76590701428679196845642516025, −16.90619923332003835512601445891, −16.50542792173258335077242087823, −15.855496828182158751252416349168, −15.004754293150773409012989621390, −14.75091826076835988162793045633, −13.71205355847010111072022139190, −13.25536113966959082789783358266, −12.10690169944300429656333745770, −11.50289067203940045150453196367, −10.81241420643248785305001169536, −9.902688907712825672054734876393, −8.92796902641370164487666528001, −8.7871579270945033036666055194, −7.58871150361471611073296857456, −7.32351690512601517544364962124, −6.47909801924013359221419708371, −5.68958537824278660571764862929, −4.68052921161319875069261205700, −4.292821696757488283886510596563, −3.25942420811553771760014681007, −2.19210934501876530609773209130, −0.83506503602778732895096718275, −0.07323690547229111144061645523, 1.26014116134069369047298592693, 2.04817436205003182589940696430, 2.989051754911573768543532305452, 3.80494107177363383181639133444, 4.38476861084327876222478234955, 5.07471348513698179463715188001, 6.352780402658346832909635241376, 7.330199677387064555847666360844, 7.81164813449120035810024371364, 8.59237828830413990781135984033, 9.24955055695409096659049019540, 10.25742933078804516415133468359, 10.52601732400285802822568219803, 11.53618347308577248566856564920, 12.15334325679424554652424802756, 12.53477189492192909724768906193, 13.24792292588436359131290023067, 14.386395784495214552820733951150, 14.9009046966200928556306881096, 15.60166976782853516382482353714, 16.656817632185667094428944632965, 17.14179412799146082563936498699, 17.74359498925273180468105561824, 18.70770963549881422581989309692, 19.163211646573088343628794726710

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