↑

Properties

Label 1-1449-1449.1229-r0-0-0
Degree $1$
Conductor $1449$
Sign $0.942 - 0.333i$
Analytic cond. $6.72912$
Root an. cond. $6.72912$
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  + (−0.928 + 0.371i)2-s + (0.723 − 0.690i)4-s + (−0.654 − 0.755i)5-s + (−0.415 + 0.909i)8-s + (0.888 + 0.458i)10-s + (−0.142 − 0.989i)11-s + (−0.888 − 0.458i)13-s + (0.0475 − 0.998i)16-s + (0.723 + 0.690i)17-s + (−0.723 + 0.690i)19-s + (−0.995 − 0.0950i)20-s + (0.5 + 0.866i)22-s + (−0.142 + 0.989i)25-s + (0.995 + 0.0950i)26-s + (−0.235 + 0.971i)29-s + ⋯
L(s)  = 1  + (−0.928 + 0.371i)2-s + (0.723 − 0.690i)4-s + (−0.654 − 0.755i)5-s + (−0.415 + 0.909i)8-s + (0.888 + 0.458i)10-s + (−0.142 − 0.989i)11-s + (−0.888 − 0.458i)13-s + (0.0475 − 0.998i)16-s + (0.723 + 0.690i)17-s + (−0.723 + 0.690i)19-s + (−0.995 − 0.0950i)20-s + (0.5 + 0.866i)22-s + (−0.142 + 0.989i)25-s + (0.995 + 0.0950i)26-s + (−0.235 + 0.971i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(6.72912\)
Root analytic conductor: \(6.72912\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1449,\ (0:\ ),\ 0.942 - 0.333i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7071269294 - 0.1213253729i\)
\(L(\frac12)\) \(\approx\) \(0.7071269294 - 0.1213253729i\)
\(L(1)\) \(\approx\) \(0.6150121168 + 0.01984913348i\)
\(L(1)\) \(\approx\) \(0.6150121168 + 0.01984913348i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.928 + 0.371i)T \)
5 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (-0.888 - 0.458i)T \)
17 \( 1 + (0.723 + 0.690i)T \)
19 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (-0.235 + 0.971i)T \)
31 \( 1 + (0.580 + 0.814i)T \)
37 \( 1 + (0.327 + 0.945i)T \)
41 \( 1 + (0.327 - 0.945i)T \)
43 \( 1 + (0.995 + 0.0950i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.0475 - 0.998i)T \)
59 \( 1 + (-0.0475 - 0.998i)T \)
61 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (0.786 + 0.618i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (0.235 + 0.971i)T \)
79 \( 1 + (0.888 + 0.458i)T \)
83 \( 1 + (-0.327 - 0.945i)T \)
89 \( 1 + (-0.995 - 0.0950i)T \)
97 \( 1 + (-0.981 + 0.189i)T \)
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

−20.613927456371859749720851143673, −19.74221926077295306749497998071, −19.284598650801938338777099313504, −18.566065542786496425589094240734, −17.8652948487011487609872226886, −17.123798135386047118557832701917, −16.39637492424211891521883502158, −15.38339654929894836819009619211, −15.03492129003027980689470429180, −14.01562314255116411030728100676, −12.814694328751777258928988555049, −12.09178696449675295320646145134, −11.499239657597032651892384965962, −10.72146346644634838541726876516, −9.86213028217638688370750855338, −9.384834891956456327658141212605, −8.222314535934445613329655820361, −7.41178189545331937292239883672, −7.09279941507532236147867631923, −6.052405498765285322574935990112, −4.58794002589411755865369763297, −3.86642399888042589189910693510, −2.56929046644658117091053653956, −2.304504177239901535431974071541, −0.68001304825097898946698937982, 0.61445566606949092655531255144, 1.552840823879356372678613584, 2.81165189842227698533518842050, 3.821939798008040882853207456264, 5.06703289003632090354713536429, 5.68532964463771094796108479658, 6.676821080410161250025115656019, 7.69775611681672073235555220427, 8.22239038513618990423106061164, 8.82105992513975099505654002261, 9.79604143000543059471210017648, 10.56797011045352747152930409787, 11.30222009126896954985973566373, 12.27306158518865700504575178044, 12.77123433622502559502647565484, 14.13766689573015355772337901708, 14.75282204727204119373901725567, 15.658678002587448328255408390735, 16.22584367335918185593409268074, 16.971278812778237719571190692165, 17.407388231163859089646978588598, 18.566012109308331560783870321588, 19.23694129568053809067452295146, 19.59547387504767386113456603097, 20.583835173218604853432506911721

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