↑

Properties

Label 1-1441-1441.560-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.684 + 0.728i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (−0.715 − 0.698i)2-s + (0.644 − 0.764i)3-s + (0.0241 + 0.999i)4-s + (−0.527 − 0.849i)5-s + (−0.995 + 0.0965i)6-s + (−0.399 + 0.916i)7-s + (0.681 − 0.732i)8-s + (−0.168 − 0.985i)9-s + (−0.215 + 0.976i)10-s + (0.779 + 0.626i)12-s + (−0.215 − 0.976i)13-s + (0.926 − 0.377i)14-s + (−0.989 − 0.144i)15-s + (−0.998 + 0.0483i)16-s + (0.262 − 0.964i)17-s + (−0.568 + 0.822i)18-s + ⋯
L(s)  = 1  + (−0.715 − 0.698i)2-s + (0.644 − 0.764i)3-s + (0.0241 + 0.999i)4-s + (−0.527 − 0.849i)5-s + (−0.995 + 0.0965i)6-s + (−0.399 + 0.916i)7-s + (0.681 − 0.732i)8-s + (−0.168 − 0.985i)9-s + (−0.215 + 0.976i)10-s + (0.779 + 0.626i)12-s + (−0.215 − 0.976i)13-s + (0.926 − 0.377i)14-s + (−0.989 − 0.144i)15-s + (−0.998 + 0.0483i)16-s + (0.262 − 0.964i)17-s + (−0.568 + 0.822i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.684 + 0.728i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (560, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5669902191 - 1.311089879i\)
\(L(\frac12)\) \(\approx\) \(-0.5669902191 - 1.311089879i\)
\(L(1)\) \(\approx\) \(0.5292839658 - 0.6504463509i\)
\(L(1)\) \(\approx\) \(0.5292839658 - 0.6504463509i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.715 - 0.698i)T \)
3 \( 1 + (0.644 - 0.764i)T \)
5 \( 1 + (-0.527 - 0.849i)T \)
7 \( 1 + (-0.399 + 0.916i)T \)
13 \( 1 + (-0.215 - 0.976i)T \)
17 \( 1 + (0.262 - 0.964i)T \)
19 \( 1 + (0.354 - 0.935i)T \)
23 \( 1 + (0.485 - 0.873i)T \)
29 \( 1 + (0.168 - 0.985i)T \)
31 \( 1 + (-0.906 - 0.421i)T \)
37 \( 1 + (0.958 + 0.285i)T \)
41 \( 1 + (0.998 + 0.0483i)T \)
43 \( 1 + (0.0724 - 0.997i)T \)
47 \( 1 + (0.885 - 0.464i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.836 + 0.548i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.0724 - 0.997i)T \)
71 \( 1 + (0.120 - 0.992i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.970 - 0.239i)T \)
83 \( 1 + (0.607 - 0.794i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.861 + 0.506i)T \)
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

−20.864642011067331855402166545695, −19.92788605240713653459420475085, −19.43258895883792907611203633517, −18.97143705584494986339404614360, −17.995798973821170794897212177375, −17.04118232798982764050711884939, −16.29695753222164590334209522876, −15.96188268399749376054648801580, −14.85884622648022469504639408597, −14.45472250483072922112561632559, −13.93313644338892555998609419536, −12.78712045342284562564539515084, −11.26393581154239286507170460561, −10.86877973812511552128379083567, −9.97647609434653723088911156684, −9.53882407454886752339616086044, −8.516756330690126850441954721801, −7.687178383544556581318044520774, −7.17979346851391345964187765483, −6.294330381418472594930414149979, −5.22211760880000770146817033460, −4.061204940920261584368769499863, −3.59704312702673316334203455791, −2.3466814099515094202420541830, −1.18923439366265842912359927392, 0.46667212029101374626362444229, 0.77101196828145273918153904169, 2.23895095593643270878042673221, 2.75614137040489865248318785768, 3.64903808382281098534499026709, 4.791027945304554487889067574866, 5.89117529828885662704777550125, 7.14227663757243289651093964326, 7.72837619402611037346056153018, 8.51178263112966412918783672168, 9.16051406499621906232370077794, 9.623411480484359582223537743605, 10.92610205292411465217674692415, 11.9507087615101308281995568938, 12.207897915508255554647364040710, 13.08048232886609803225880776525, 13.4926764937693745785176004651, 14.92074020610394236924804957435, 15.57966721758048868498337082092, 16.382327785498193245607862095601, 17.26858090553985779934987968543, 18.10705156536633498261733649731, 18.63824622696507464156695329202, 19.37839183594029281793899756443, 19.990636623800519399470473702877

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