Properties

Label 1-1441-1441.1109-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.640 + 0.767i$
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  + (0.309 + 0.951i)2-s + 3-s + (−0.809 + 0.587i)4-s + 5-s + (0.309 + 0.951i)6-s + (0.309 − 0.951i)7-s + (−0.809 − 0.587i)8-s + 9-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)12-s + (−0.809 + 0.587i)13-s + 14-s + 15-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + 3-s + (−0.809 + 0.587i)4-s + 5-s + (0.309 + 0.951i)6-s + (0.309 − 0.951i)7-s + (−0.809 − 0.587i)8-s + 9-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)12-s + (−0.809 + 0.587i)13-s + 14-s + 15-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.640 + 0.767i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.640 + 0.767i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.828144272 + 1.323835490i\)
\(L(\frac12)\) \(\approx\) \(2.828144272 + 1.323835490i\)
\(L(1)\) \(\approx\) \(1.792239459 + 0.7334273152i\)
\(L(1)\) \(\approx\) \(1.792239459 + 0.7334273152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + T \)
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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

−20.7603244674183629101151057550, −20.0585075917447191036887586548, −19.140202988890850375303998429, −18.528790664361814867615144204750, −17.98221748197019817596041062005, −17.12200136020251457134493096408, −15.7906194741752539703043131193, −14.990253660131811312824306188177, −14.35149699730154706672196857478, −13.79408354897275952246686000330, −13.01389138671951313273090703416, −12.29350395483400964275081084813, −11.61069581445935286946010517426, −10.26863518774483235383372887847, −9.904651493096518249679879851493, −9.15226104471400047947574934699, −8.49511327801333311374104320485, −7.50860557884422740499452415809, −6.24078987157397856716516429904, −5.21225576890652845732627833962, −4.8281901290001099271622510579, −3.288072981722512480005496720330, −2.87622915999810856757101614579, −1.98087211089159745604827733646, −1.293998170478932301671442870968, 1.07580621457588962578479645898, 2.29806740076989441384910426383, 3.175286681088375564145514370932, 4.39080280142869547953336268714, 4.698250044937603605895677535845, 6.00578534874545801407263962202, 6.843606688977327529320980091007, 7.41672450660162135875317136771, 8.34751401018500168233430740978, 9.014433389620000859060165800695, 9.89516897464610383419686018984, 10.394709859086763258965560150943, 11.9029435701276357629031188238, 12.96387737423096976055465586170, 13.45341137200834170245602408983, 14.06293996861202013033589919090, 14.68648670352136933519009480060, 15.24646749878071483495181349954, 16.42003701353933040930496740692, 16.98025934179094415592594944574, 17.63540304452103823490746554554, 18.43668052229070924465980561906, 19.32965036292891402448244873328, 20.12514080506299397509462327356, 21.11091190466725051576182062296

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