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Properties

Label 1-1441-1441.630-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.979 + 0.202i$
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  + (0.998 + 0.0483i)2-s + (−0.607 + 0.794i)3-s + (0.995 + 0.0965i)4-s + (0.0241 − 0.999i)5-s + (−0.644 + 0.764i)6-s + (−0.970 − 0.239i)7-s + (0.989 + 0.144i)8-s + (−0.262 − 0.964i)9-s + (0.0724 − 0.997i)10-s + (−0.681 + 0.732i)12-s + (−0.970 − 0.239i)13-s + (−0.958 − 0.285i)14-s + (0.779 + 0.626i)15-s + (0.981 + 0.192i)16-s + (0.681 + 0.732i)17-s + (−0.215 − 0.976i)18-s + ⋯
L(s)  = 1  + (0.998 + 0.0483i)2-s + (−0.607 + 0.794i)3-s + (0.995 + 0.0965i)4-s + (0.0241 − 0.999i)5-s + (−0.644 + 0.764i)6-s + (−0.970 − 0.239i)7-s + (0.989 + 0.144i)8-s + (−0.262 − 0.964i)9-s + (0.0724 − 0.997i)10-s + (−0.681 + 0.732i)12-s + (−0.970 − 0.239i)13-s + (−0.958 − 0.285i)14-s + (0.779 + 0.626i)15-s + (0.981 + 0.192i)16-s + (0.681 + 0.732i)17-s + (−0.215 − 0.976i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.979 + 0.202i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (630, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.979 + 0.202i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001584546314 + 0.01550375844i\)
\(L(\frac12)\) \(\approx\) \(0.001584546314 + 0.01550375844i\)
\(L(1)\) \(\approx\) \(1.238147802 + 0.005996439144i\)
\(L(1)\) \(\approx\) \(1.238147802 + 0.005996439144i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.998 + 0.0483i)T \)
3 \( 1 + (-0.607 + 0.794i)T \)
5 \( 1 + (0.0241 - 0.999i)T \)
7 \( 1 + (-0.970 - 0.239i)T \)
13 \( 1 + (-0.970 - 0.239i)T \)
17 \( 1 + (0.681 + 0.732i)T \)
19 \( 1 + (-0.485 - 0.873i)T \)
23 \( 1 + (0.168 - 0.985i)T \)
29 \( 1 + (0.354 + 0.935i)T \)
31 \( 1 + (0.443 - 0.896i)T \)
37 \( 1 + (0.748 + 0.663i)T \)
41 \( 1 + (-0.906 - 0.421i)T \)
43 \( 1 + (0.0241 - 0.999i)T \)
47 \( 1 + (0.262 + 0.964i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.681 - 0.732i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.0241 - 0.999i)T \)
71 \( 1 + (-0.715 - 0.698i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.958 + 0.285i)T \)
83 \( 1 + (-0.995 + 0.0965i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.926 + 0.377i)T \)
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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

−21.45493507676805605929957764510, −20.00303750985534099158045456843, −19.32662204642460017195337876767, −18.93015355120889388060105591242, −18.028851640058977667533294997266, −17.020226019832457615035684962592, −16.39203471064683830180583866312, −15.55470202962891033461963364875, −14.69886078344610062678168016974, −13.99140730135193956534736142092, −13.332302195706180049672959680881, −12.48439068494276641452929462375, −11.89334299527729202723430919553, −11.31715054959238623195654186500, −10.22374067072850378096681642726, −9.79640346379023321264585654034, −8.04623671548035776626166568593, −7.21327882106019882802853328863, −6.75622328643095528643630922054, −5.94606810566699145003308115122, −5.36003610407613050604224164860, −4.17087412346927132248355127080, −3.04648763754743629355544508513, −2.5499165650879324408314246885, −1.490882972426179219255328676840, 0.00224638700793261618632395934, 1.05609576555753754287375416727, 2.554925242669730048470709463316, 3.44118133489335014016386324653, 4.40048951661528482535066336610, 4.827196809464415102341296265153, 5.7791498693841787995212831970, 6.38600349162390010463785367373, 7.34705340163744308776095166948, 8.51299103242279728759606191631, 9.48743895675439214433723083983, 10.24097460307457249325471797589, 10.8900007185465659327425929532, 12.100872815501461439233311387393, 12.39282432507308949434801567575, 13.09752122854325797529475382890, 14.02342871211038109156725589984, 15.08322035252283051354279465902, 15.479745756797510565707225646159, 16.415619612641864174221998371263, 16.8943835951825994032274343180, 17.28191092448731079770330753518, 18.83691865472429224417398010135, 19.863414493446268230952379683583, 20.20381928287211466478872894870

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