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Properties

Label 1-1441-1441.206-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.890 - 0.455i$
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  + (−0.926 − 0.377i)2-s + (0.120 + 0.992i)3-s + (0.715 + 0.698i)4-s + (0.120 + 0.992i)5-s + (0.262 − 0.964i)6-s + (0.262 + 0.964i)7-s + (−0.399 − 0.916i)8-s + (−0.970 + 0.239i)9-s + (0.262 − 0.964i)10-s + (−0.607 + 0.794i)12-s + (0.998 + 0.0483i)13-s + (0.120 − 0.992i)14-s + (−0.970 + 0.239i)15-s + (0.0241 + 0.999i)16-s + (0.943 − 0.331i)17-s + (0.989 + 0.144i)18-s + ⋯
L(s)  = 1  + (−0.926 − 0.377i)2-s + (0.120 + 0.992i)3-s + (0.715 + 0.698i)4-s + (0.120 + 0.992i)5-s + (0.262 − 0.964i)6-s + (0.262 + 0.964i)7-s + (−0.399 − 0.916i)8-s + (−0.970 + 0.239i)9-s + (0.262 − 0.964i)10-s + (−0.607 + 0.794i)12-s + (0.998 + 0.0483i)13-s + (0.120 − 0.992i)14-s + (−0.970 + 0.239i)15-s + (0.0241 + 0.999i)16-s + (0.943 − 0.331i)17-s + (0.989 + 0.144i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.890 - 0.455i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.890 - 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2659498171 + 1.103047382i\)
\(L(\frac12)\) \(\approx\) \(-0.2659498171 + 1.103047382i\)
\(L(1)\) \(\approx\) \(0.6508659367 + 0.4571429072i\)
\(L(1)\) \(\approx\) \(0.6508659367 + 0.4571429072i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.926 - 0.377i)T \)
3 \( 1 + (0.120 + 0.992i)T \)
5 \( 1 + (0.120 + 0.992i)T \)
7 \( 1 + (0.262 + 0.964i)T \)
13 \( 1 + (0.998 + 0.0483i)T \)
17 \( 1 + (0.943 - 0.331i)T \)
19 \( 1 + (0.943 + 0.331i)T \)
23 \( 1 + (0.995 + 0.0965i)T \)
29 \( 1 + (-0.926 + 0.377i)T \)
31 \( 1 + (-0.748 + 0.663i)T \)
37 \( 1 + (-0.443 + 0.896i)T \)
41 \( 1 + (0.607 - 0.794i)T \)
43 \( 1 + (0.906 - 0.421i)T \)
47 \( 1 + (0.644 - 0.764i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.958 + 0.285i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.906 - 0.421i)T \)
71 \( 1 + (-0.861 - 0.506i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.981 - 0.192i)T \)
83 \( 1 + (-0.885 - 0.464i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.354 + 0.935i)T \)
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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

−20.0757044403184009177199342235, −19.1992301450026933406576041437, −18.63167389880036123753307818096, −17.65140891482111660152503396619, −17.29878150135167829588927321016, −16.50610990182047023600702892150, −15.93157210197467670189193635786, −14.70229227566157635862814713296, −14.050860952493989211997025088711, −13.21024235650413479590664225349, −12.57778996802827414347632525143, −11.401030985905544472033539822741, −11.026976032798808093097197243764, −9.77528906091981408311753793438, −9.08651962267277804549343901671, −8.29271023442172097528786331143, −7.61329500434002461291160879019, −7.09791619527593138612105046278, −5.893102098934297286890038632928, −5.47199019939060906805119017731, −4.086413926239878912622626580206, −2.86274783370959800107087422902, −1.48952125919396930466685397058, −1.16569762888669884949983477467, −0.31155423689663866766656918246, 1.31386297993102331209367574321, 2.42641534099085328841052446502, 3.23405246738967509212511742793, 3.71191645696811807519722923111, 5.30900025252626613073378082425, 5.91586297321632069678872831098, 7.07474168898943682673077902302, 7.89171198315545304258789262174, 8.882388318380379846130402507086, 9.29700191558692105220083615966, 10.23481420074161493644699506431, 10.825120847844550822703560883348, 11.50789782316649379237974980512, 12.12040585859586587653686180087, 13.41037617276644156712488675886, 14.45718901476194488508516726400, 15.02144740464049039353845517117, 15.87539043031986412775770290516, 16.30038073391902368669834735183, 17.36907686997281310459502558493, 18.06284835840298312472575632254, 18.78148259930961324638310863234, 19.195668918393753647332463495751, 20.45296032435347101523688963430, 20.86802532160449376275106281884

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