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Properties

Label 1-1441-1441.1015-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.836 - 0.548i$
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  + (−0.215 + 0.976i)2-s + (−0.262 + 0.964i)3-s + (−0.906 − 0.421i)4-s + (−0.998 + 0.0483i)5-s + (−0.885 − 0.464i)6-s + (0.715 + 0.698i)7-s + (0.607 − 0.794i)8-s + (−0.861 − 0.506i)9-s + (0.168 − 0.985i)10-s + (0.644 − 0.764i)12-s + (−0.989 − 0.144i)13-s + (−0.836 + 0.548i)14-s + (0.215 − 0.976i)15-s + (0.644 + 0.764i)16-s + (0.970 + 0.239i)17-s + (0.681 − 0.732i)18-s + ⋯
L(s)  = 1  + (−0.215 + 0.976i)2-s + (−0.262 + 0.964i)3-s + (−0.906 − 0.421i)4-s + (−0.998 + 0.0483i)5-s + (−0.885 − 0.464i)6-s + (0.715 + 0.698i)7-s + (0.607 − 0.794i)8-s + (−0.861 − 0.506i)9-s + (0.168 − 0.985i)10-s + (0.644 − 0.764i)12-s + (−0.989 − 0.144i)13-s + (−0.836 + 0.548i)14-s + (0.215 − 0.976i)15-s + (0.644 + 0.764i)16-s + (0.970 + 0.239i)17-s + (0.681 − 0.732i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.836 - 0.548i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.836 - 0.548i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3702175496 + 0.1105071617i\)
\(L(\frac12)\) \(\approx\) \(-0.3702175496 + 0.1105071617i\)
\(L(1)\) \(\approx\) \(0.3398129441 + 0.5138656180i\)
\(L(1)\) \(\approx\) \(0.3398129441 + 0.5138656180i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.215 + 0.976i)T \)
3 \( 1 + (-0.262 + 0.964i)T \)
5 \( 1 + (-0.998 + 0.0483i)T \)
7 \( 1 + (0.715 + 0.698i)T \)
13 \( 1 + (-0.989 - 0.144i)T \)
17 \( 1 + (0.970 + 0.239i)T \)
19 \( 1 + (-0.644 + 0.764i)T \)
23 \( 1 + (0.607 + 0.794i)T \)
29 \( 1 + (-0.399 + 0.916i)T \)
31 \( 1 + (0.607 - 0.794i)T \)
37 \( 1 + (-0.981 + 0.192i)T \)
41 \( 1 + (-0.970 + 0.239i)T \)
43 \( 1 + (0.836 + 0.548i)T \)
47 \( 1 + (-0.215 + 0.976i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.527 + 0.849i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.836 + 0.548i)T \)
71 \( 1 + (0.607 - 0.794i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.998 - 0.0483i)T \)
83 \( 1 + (-0.981 - 0.192i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.715 - 0.698i)T \)
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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

−19.690261554731097847432076291267, −19.1459159355771786977978945631, −18.66959066744965753998125517935, −17.60570248485810267100218020658, −17.14249017501346625096372862533, −16.49049905361721618837498277337, −15.06887140085487563352974200250, −14.285315245434983388538348612823, −13.63674301209673591172487385678, −12.62485246922434305661465822479, −12.17831829535410408072328150569, −11.45737799097003535491705883491, −10.859885828210710354930116120528, −10.05149756759963425352308083017, −8.75720611012444481758262122363, −8.17158372967916421279894842215, −7.41907190358496009291065889425, −6.83619649377596230237121543753, −5.168761461113209417560802846853, −4.68819138808213624957541890978, −3.61782565104661729119244550623, −2.658651270946444277857623783214, −1.72200137995188508710524786144, −0.67081414423620733656291755889, −0.13588261140304120491514952867, 1.27215803316518085703955746468, 2.97520290746945538964557044208, 3.91842894392159388611601446579, 4.749722173696901846254026701662, 5.31646912365344012025349572495, 6.132406968794715627093559870548, 7.32891167609767108381135149355, 8.00685355241264134735145000726, 8.68148814588071531887183272245, 9.50417888636619733070781687945, 10.31999669356058636935282047085, 11.14830980834159685705989916473, 12.03698956360829117528443538575, 12.67392510467625569567225735850, 14.20976624652595022202864710307, 14.70486543028658540823684038383, 15.244138678618311205946898567433, 15.78971523621994263908242281494, 16.75724035546235965385930215740, 17.113991847670663236993420198307, 18.050487701500536721261871591201, 18.973443760729436957357530854694, 19.45122270949177489136151901892, 20.61008376350630826616272856639, 21.30116851901575612322516653312

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