Properties

Label 1-1441-1441.493-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.882 + 0.470i$
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  + (−0.443 + 0.896i)2-s + (0.644 − 0.764i)3-s + (−0.607 − 0.794i)4-s + (−0.527 − 0.849i)5-s + (0.399 + 0.916i)6-s + (−0.748 − 0.663i)7-s + (0.981 − 0.192i)8-s + (−0.168 − 0.985i)9-s + (0.995 − 0.0965i)10-s + (−0.998 − 0.0483i)12-s + (−0.748 − 0.663i)13-s + (0.926 − 0.377i)14-s + (−0.989 − 0.144i)15-s + (−0.262 + 0.964i)16-s + (−0.998 + 0.0483i)17-s + (0.958 + 0.285i)18-s + ⋯
L(s)  = 1  + (−0.443 + 0.896i)2-s + (0.644 − 0.764i)3-s + (−0.607 − 0.794i)4-s + (−0.527 − 0.849i)5-s + (0.399 + 0.916i)6-s + (−0.748 − 0.663i)7-s + (0.981 − 0.192i)8-s + (−0.168 − 0.985i)9-s + (0.995 − 0.0965i)10-s + (−0.998 − 0.0483i)12-s + (−0.748 − 0.663i)13-s + (0.926 − 0.377i)14-s + (−0.989 − 0.144i)15-s + (−0.262 + 0.964i)16-s + (−0.998 + 0.0483i)17-s + (0.958 + 0.285i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.882 + 0.470i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.882 + 0.470i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07946756996 - 0.3177962460i\)
\(L(\frac12)\) \(\approx\) \(-0.07946756996 - 0.3177962460i\)
\(L(1)\) \(\approx\) \(0.6317055076 - 0.1917867962i\)
\(L(1)\) \(\approx\) \(0.6317055076 - 0.1917867962i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.443 - 0.896i)T \)
3 \( 1 + (-0.644 + 0.764i)T \)
5 \( 1 + (0.527 + 0.849i)T \)
7 \( 1 + (0.748 + 0.663i)T \)
13 \( 1 + (0.748 + 0.663i)T \)
17 \( 1 + (0.998 - 0.0483i)T \)
19 \( 1 + (-0.779 - 0.626i)T \)
23 \( 1 + (0.681 + 0.732i)T \)
29 \( 1 + (-0.885 - 0.464i)T \)
31 \( 1 + (0.906 + 0.421i)T \)
37 \( 1 + (-0.568 + 0.822i)T \)
41 \( 1 + (-0.836 + 0.548i)T \)
43 \( 1 + (0.527 + 0.849i)T \)
47 \( 1 + (0.168 + 0.985i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.998 - 0.0483i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.527 - 0.849i)T \)
71 \( 1 + (-0.485 - 0.873i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.926 - 0.377i)T \)
83 \( 1 + (0.607 - 0.794i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.861 - 0.506i)T \)
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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

−21.299896083869120017751646419415, −20.063431327467326786942402304327, −19.67614892230769262036144876899, −19.23710999061291618312752780658, −18.32828525588401422771373407263, −17.685244396786418853041815072932, −16.47862125739254376001356482463, −15.8900215022253712157369457264, −15.18929154524330791053082838047, −14.25865116909211507448618786441, −13.59577385356362165181141932017, −12.66423286206078432449062885553, −11.63616306118411681119981711, −11.253674812281962768406550257595, −10.2361367402770491345339055708, −9.5658154915391573852745880220, −9.11463907266158626212899806782, −8.095846795273985063612952375085, −7.3757544762851087395478929464, −6.345870610127281141510438881096, −4.88141363443261757421027351410, −4.20313696464185781321897402547, −3.17116156441031055346203364306, −2.791235289511507972113414026693, −1.9093173654970992878221149273, 0.15323292617770560525230345501, 0.98789551763658878459586743864, 2.19272853129485796124838667852, 3.55055347220829028940839833070, 4.32471197755272090521810257802, 5.40346787426377476692476551304, 6.33842809312055523510693839585, 7.186893024121060544742016552305, 7.724435936962495645601523073497, 8.44127029721994609775670772965, 9.23508719311521840369852608057, 9.86759348976614435840526991786, 10.89536648661085376816971398108, 12.280053807936666880470092277649, 12.736605627465393175819583303815, 13.56119463982496145533424298775, 14.19558157049330949016060321611, 15.08920474571272891738233394230, 15.84302698251551563489276559758, 16.48445346588503935610961025481, 17.263611850376322522505892498911, 17.99097894622551801698449958128, 18.75350120915709122208985698724, 19.69889758712799121619969687229, 19.9470094046061832606016723025

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