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Properties

Label 1-1441-1441.1040-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.698 - 0.715i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.885 + 0.464i)2-s + (0.644 − 0.764i)3-s + (0.568 − 0.822i)4-s + (−0.527 − 0.849i)5-s + (−0.215 + 0.976i)6-s + (−0.995 − 0.0965i)7-s + (−0.120 + 0.992i)8-s + (−0.168 − 0.985i)9-s + (0.861 + 0.506i)10-s + (−0.262 − 0.964i)12-s + (−0.399 + 0.916i)13-s + (0.926 − 0.377i)14-s + (−0.989 − 0.144i)15-s + (−0.354 − 0.935i)16-s + (−0.836 − 0.548i)17-s + (0.607 + 0.794i)18-s + ⋯
L(s)  = 1  + (−0.885 + 0.464i)2-s + (0.644 − 0.764i)3-s + (0.568 − 0.822i)4-s + (−0.527 − 0.849i)5-s + (−0.215 + 0.976i)6-s + (−0.995 − 0.0965i)7-s + (−0.120 + 0.992i)8-s + (−0.168 − 0.985i)9-s + (0.861 + 0.506i)10-s + (−0.262 − 0.964i)12-s + (−0.399 + 0.916i)13-s + (0.926 − 0.377i)14-s + (−0.989 − 0.144i)15-s + (−0.354 − 0.935i)16-s + (−0.836 − 0.548i)17-s + (0.607 + 0.794i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.698 - 0.715i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1040, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.698 - 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9516719073 - 0.4007742163i\)
\(L(\frac12)\) \(\approx\) \(0.9516719073 - 0.4007742163i\)
\(L(1)\) \(\approx\) \(0.6739316253 - 0.1643020501i\)
\(L(1)\) \(\approx\) \(0.6739316253 - 0.1643020501i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.885 + 0.464i)T \)
3 \( 1 + (0.644 - 0.764i)T \)
5 \( 1 + (-0.527 - 0.849i)T \)
7 \( 1 + (-0.995 - 0.0965i)T \)
13 \( 1 + (-0.399 + 0.916i)T \)
17 \( 1 + (-0.836 - 0.548i)T \)
19 \( 1 + (0.998 + 0.0483i)T \)
23 \( 1 + (0.981 + 0.192i)T \)
29 \( 1 + (0.989 - 0.144i)T \)
31 \( 1 + (-0.906 - 0.421i)T \)
37 \( 1 + (0.0241 + 0.999i)T \)
41 \( 1 + (-0.779 - 0.626i)T \)
43 \( 1 + (-0.644 + 0.764i)T \)
47 \( 1 + (0.715 + 0.698i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.354 - 0.935i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.644 + 0.764i)T \)
71 \( 1 + (-0.681 + 0.732i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.644 + 0.764i)T \)
83 \( 1 + (0.607 - 0.794i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.861 + 0.506i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.24175772203164063561867613681, −19.820121297595154213945573450447, −19.37486242608831485635106923842, −18.50743658219455859546955719963, −17.821303606033182848941159413835, −16.7658887751454808245095975510, −16.09156290135314871802327076245, −15.37309815815969104620526955655, −14.976869398662381766278585535496, −13.73758660470523624640403064658, −12.93263612831230710182793679190, −12.04875157541255976535825198887, −11.09340637648802398198476394086, −10.38496950779596136540092206468, −10.00884479674413274078919264218, −8.993346619581560259808568561770, −8.457884564159228591098696765, −7.36281295184978310658657046395, −6.942290779150346426689671939716, −5.66442551835675481584803379080, −4.313493195841234892732829001089, −3.30574352989891009895436378784, −3.051609304087640323161223708098, −2.12252102192190415947788655565, −0.47799506316233440934682379380, 0.51268104182320380674282352787, 1.30327686540043562267159476778, 2.39878217103698317412719652319, 3.33930108870676425324713026773, 4.57375688788675339230990499562, 5.622949604469848741463887823330, 6.7981168933798744135632883448, 7.05492679849869017424854697775, 7.987315343123653188792198230847, 8.86764781924651151931344853429, 9.273633576845397072969000967302, 9.982862749737574132982553396536, 11.42863432832183051996583032427, 11.89726708786115437698130010726, 12.90736344021315951265166736021, 13.589598353097676292767409726921, 14.409502517861900744675212326130, 15.43950122103577965498625927209, 15.908860645978776651020345269736, 16.742929897323660998606503456054, 17.36359498738089660223449571174, 18.365839808443787976134758445301, 19.025585099365096305774323679858, 19.50874130523765626980521075773, 20.25759126477884373249397579604

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