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Properties

Label 1-1441-1441.255-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.657 - 0.753i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (−0.527 + 0.849i)2-s + (0.981 − 0.192i)3-s + (−0.443 − 0.896i)4-s + (−0.681 + 0.732i)5-s + (−0.354 + 0.935i)6-s + (−0.836 − 0.548i)7-s + (0.995 + 0.0965i)8-s + (0.926 − 0.377i)9-s + (−0.262 − 0.964i)10-s + (−0.607 − 0.794i)12-s + (−0.779 + 0.626i)13-s + (0.906 − 0.421i)14-s + (−0.527 + 0.849i)15-s + (−0.607 + 0.794i)16-s + (0.568 + 0.822i)17-s + (−0.168 + 0.985i)18-s + ⋯
L(s)  = 1  + (−0.527 + 0.849i)2-s + (0.981 − 0.192i)3-s + (−0.443 − 0.896i)4-s + (−0.681 + 0.732i)5-s + (−0.354 + 0.935i)6-s + (−0.836 − 0.548i)7-s + (0.995 + 0.0965i)8-s + (0.926 − 0.377i)9-s + (−0.262 − 0.964i)10-s + (−0.607 − 0.794i)12-s + (−0.779 + 0.626i)13-s + (0.906 − 0.421i)14-s + (−0.527 + 0.849i)15-s + (−0.607 + 0.794i)16-s + (0.568 + 0.822i)17-s + (−0.168 + 0.985i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.657 - 0.753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6636068364 - 0.3016587600i\)
\(L(\frac12)\) \(\approx\) \(0.6636068364 - 0.3016587600i\)
\(L(1)\) \(\approx\) \(0.7634148437 + 0.1735423539i\)
\(L(1)\) \(\approx\) \(0.7634148437 + 0.1735423539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.527 + 0.849i)T \)
3 \( 1 + (0.981 - 0.192i)T \)
5 \( 1 + (-0.681 + 0.732i)T \)
7 \( 1 + (-0.836 - 0.548i)T \)
13 \( 1 + (-0.779 + 0.626i)T \)
17 \( 1 + (0.568 + 0.822i)T \)
19 \( 1 + (-0.607 - 0.794i)T \)
23 \( 1 + (-0.995 + 0.0965i)T \)
29 \( 1 + (0.644 - 0.764i)T \)
31 \( 1 + (-0.995 - 0.0965i)T \)
37 \( 1 + (0.989 - 0.144i)T \)
41 \( 1 + (-0.568 + 0.822i)T \)
43 \( 1 + (0.906 + 0.421i)T \)
47 \( 1 + (0.527 - 0.849i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.0241 - 0.999i)T \)
61 \( 1 - T \)
67 \( 1 + (0.906 - 0.421i)T \)
71 \( 1 + (-0.995 - 0.0965i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.681 + 0.732i)T \)
83 \( 1 + (-0.989 - 0.144i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.836 - 0.548i)T \)
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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

−20.40224417922671317517400338645, −20.18370170501901491077224204911, −19.4225088188550239176923350326, −18.84489961000003294857263826009, −18.193235085678187129695276287339, −16.985009630692394854109670211958, −16.24000155288217070349806395651, −15.76465367058054696562974937426, −14.737423934090603121886983827470, −13.87079554861650314055714504259, −12.85541945228275742768801876963, −12.45709774579795952100170158948, −11.87633805886332129428671155297, −10.54615681445873994233759486420, −9.93852461703107889953051576988, −9.14904407421299690422760596784, −8.65509712849435118435947266893, −7.75443360024588150123340093474, −7.261179265838693159064167084606, −5.623578415301496805872264401543, −4.5458235523553398356397838573, −3.79170534604912319574147004451, −2.99920638376209263795397574422, −2.28723236777611867135249433118, −1.08675377535904141485227406629, 0.321645183600019713948948281141, 1.82313657482402180091188241006, 2.83674803813337155781540130186, 3.93229004086887116658385036915, 4.44375533701903068717890965783, 6.07038744831456147430292586422, 6.731056515180632042705804186583, 7.39868517774926647935022688232, 7.97441384339666040544644161639, 8.834216004911554446967543049820, 9.80807863859455897390570179786, 10.155418084470966893676124038531, 11.2128333812179526752556091022, 12.41683257057804018770335999225, 13.25103637647783210105670575143, 14.0893369912138001366227169700, 14.65215541306984548433905810928, 15.28028641835952432188840055771, 16.01434314310416844263023293561, 16.72839058470293835856074962720, 17.64643020206449310150618445997, 18.57892632527762834502271223277, 19.105196374854501315549171694497, 19.773454578460135200034550818960, 19.98665504805146959969927069590

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