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Properties

Label 1-1441-1441.490-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.996 - 0.0874i$
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  + (−0.0724 − 0.997i)2-s + (−0.906 − 0.421i)3-s + (−0.989 + 0.144i)4-s + (0.485 + 0.873i)5-s + (−0.354 + 0.935i)6-s + (0.262 − 0.964i)7-s + (0.215 + 0.976i)8-s + (0.644 + 0.764i)9-s + (0.836 − 0.548i)10-s + (0.958 + 0.285i)12-s + (0.998 − 0.0483i)13-s + (−0.981 − 0.192i)14-s + (−0.0724 − 0.997i)15-s + (0.958 − 0.285i)16-s + (0.568 + 0.822i)17-s + (0.715 − 0.698i)18-s + ⋯
L(s)  = 1  + (−0.0724 − 0.997i)2-s + (−0.906 − 0.421i)3-s + (−0.989 + 0.144i)4-s + (0.485 + 0.873i)5-s + (−0.354 + 0.935i)6-s + (0.262 − 0.964i)7-s + (0.215 + 0.976i)8-s + (0.644 + 0.764i)9-s + (0.836 − 0.548i)10-s + (0.958 + 0.285i)12-s + (0.998 − 0.0483i)13-s + (−0.981 − 0.192i)14-s + (−0.0724 − 0.997i)15-s + (0.958 − 0.285i)16-s + (0.568 + 0.822i)17-s + (0.715 − 0.698i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.996 - 0.0874i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (490, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.996 - 0.0874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.082420524 - 0.04741885793i\)
\(L(\frac12)\) \(\approx\) \(1.082420524 - 0.04741885793i\)
\(L(1)\) \(\approx\) \(0.8014549732 - 0.2836895028i\)
\(L(1)\) \(\approx\) \(0.8014549732 - 0.2836895028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.0724 - 0.997i)T \)
3 \( 1 + (-0.906 - 0.421i)T \)
5 \( 1 + (0.485 + 0.873i)T \)
7 \( 1 + (0.262 - 0.964i)T \)
13 \( 1 + (0.998 - 0.0483i)T \)
17 \( 1 + (0.568 + 0.822i)T \)
19 \( 1 + (0.958 + 0.285i)T \)
23 \( 1 + (-0.215 + 0.976i)T \)
29 \( 1 + (0.926 + 0.377i)T \)
31 \( 1 + (-0.215 - 0.976i)T \)
37 \( 1 + (0.443 + 0.896i)T \)
41 \( 1 + (-0.568 + 0.822i)T \)
43 \( 1 + (-0.981 + 0.192i)T \)
47 \( 1 + (0.0724 + 0.997i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.943 - 0.331i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.981 - 0.192i)T \)
71 \( 1 + (-0.215 - 0.976i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.485 + 0.873i)T \)
83 \( 1 + (-0.443 + 0.896i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.262 - 0.964i)T \)
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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

−21.05524355340516473111810942665, −20.12450764975159239301818739001, −18.74489522752289631959309349201, −18.12604073885650572190110914925, −17.76487091679977510569142043669, −16.7931890779296354677966718836, −16.05272236816982867490924306528, −15.91317457462805932871340087340, −14.87331106358296957909494965164, −13.961118997862013246491312206077, −13.20990142491407959776958476092, −12.236989093718210797602255569107, −11.81078801049867591461449297225, −10.51110016150327369503194877974, −9.76149592711573010394027542986, −8.93403321471518967443161686010, −8.51254293431207566003698169078, −7.29972636786275009634959498395, −6.29855471554926246564027833032, −5.70039785077649102522609690844, −5.07861419431810281251564796835, −4.48276291468159157694212292570, −3.2658955721117960451955509424, −1.576162296065471104820711814850, −0.56474713623289677691039205181, 1.26042356849255907229023187758, 1.518203426649739310616017136341, 2.980648792062493858574724654432, 3.747933732847630450281258766328, 4.72645853623251815740826268728, 5.741294072277451879379429926344, 6.392281297577188775921866431072, 7.55044226899566566043963086218, 8.06794901913112328702217224230, 9.54929719836585999598317834575, 10.20310694118159072161351482295, 10.784849809131560289998278529707, 11.38577828870376348786501960800, 12.0737699580934583889905600486, 13.20048736699250049561569469733, 13.59052775683550120964487604298, 14.2573476606039216239083001280, 15.36903293767680787857440628661, 16.647298089714927131100836235741, 17.113183983204506505459217280703, 18.0978857877252698761560247748, 18.22557678744274396994544406337, 19.134748107459126396538042438692, 19.90719210588112688869171812205, 20.798344508628375934706719046013

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