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Properties

Label 1-1441-1441.142-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.543 - 0.839i$
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  + (0.906 − 0.421i)2-s + (0.995 + 0.0965i)3-s + (0.644 − 0.764i)4-s + (0.399 − 0.916i)5-s + (0.943 − 0.331i)6-s + (0.607 − 0.794i)7-s + (0.262 − 0.964i)8-s + (0.981 + 0.192i)9-s + (−0.0241 − 0.999i)10-s + (0.715 − 0.698i)12-s + (−0.0241 + 0.999i)13-s + (0.215 − 0.976i)14-s + (0.485 − 0.873i)15-s + (−0.168 − 0.985i)16-s + (0.989 + 0.144i)17-s + (0.970 − 0.239i)18-s + ⋯
L(s)  = 1  + (0.906 − 0.421i)2-s + (0.995 + 0.0965i)3-s + (0.644 − 0.764i)4-s + (0.399 − 0.916i)5-s + (0.943 − 0.331i)6-s + (0.607 − 0.794i)7-s + (0.262 − 0.964i)8-s + (0.981 + 0.192i)9-s + (−0.0241 − 0.999i)10-s + (0.715 − 0.698i)12-s + (−0.0241 + 0.999i)13-s + (0.215 − 0.976i)14-s + (0.485 − 0.873i)15-s + (−0.168 − 0.985i)16-s + (0.989 + 0.144i)17-s + (0.970 − 0.239i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.543 - 0.839i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.543 - 0.839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.658301148 - 6.724654993i\)
\(L(\frac12)\) \(\approx\) \(3.658301148 - 6.724654993i\)
\(L(1)\) \(\approx\) \(2.526609925 - 1.678288309i\)
\(L(1)\) \(\approx\) \(2.526609925 - 1.678288309i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.906 - 0.421i)T \)
3 \( 1 + (0.995 + 0.0965i)T \)
5 \( 1 + (0.399 - 0.916i)T \)
7 \( 1 + (0.607 - 0.794i)T \)
13 \( 1 + (-0.0241 + 0.999i)T \)
17 \( 1 + (0.989 + 0.144i)T \)
19 \( 1 + (-0.885 - 0.464i)T \)
23 \( 1 + (0.836 - 0.548i)T \)
29 \( 1 + (-0.981 + 0.192i)T \)
31 \( 1 + (-0.998 + 0.0483i)T \)
37 \( 1 + (-0.527 - 0.849i)T \)
41 \( 1 + (0.168 - 0.985i)T \)
43 \( 1 + (0.861 - 0.506i)T \)
47 \( 1 + (0.120 + 0.992i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.443 + 0.896i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.861 - 0.506i)T \)
71 \( 1 + (-0.354 - 0.935i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.748 + 0.663i)T \)
83 \( 1 + (0.0724 + 0.997i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.958 - 0.285i)T \)
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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

−20.97188490045234035949921213611, −20.348300209598338038094780327021, −19.23304086544596812637997118702, −18.61261578986632353671114611182, −17.8281137254881016570466622174, −17.00087721575848317816555964541, −15.88091003837194801981006360859, −15.00343913126992886952246254895, −14.82695113168424826612680273667, −14.219964929286295081710208324166, −13.204209282824666572688245556055, −12.76771287085002321027508669502, −11.742345036312942045575989219491, −10.89661208812380494119404353252, −10.00458475378938477503883118345, −8.98933472019084538295598309288, −8.02009365160285301920707841383, −7.584789708189562124579126145836, −6.63739534094637480667597176744, −5.71617664963063146788574272225, −5.058169291170336467948740937949, −3.74444102762962026126428296523, −3.15579216255323217198735485239, −2.36452505649135759621276358516, −1.60104079472550897144984571607, 0.80756084207088608608205674249, 1.68477009562720705681601039086, 2.298125171940567608517769127296, 3.593937594571759601332770341107, 4.236652186528121445276535503, 4.857800818112385910619918423725, 5.81599899203998761611123875336, 7.03489189134177703433016799033, 7.62303092048792712751127660696, 8.89002735825258673876796223476, 9.29938427815713215887304807417, 10.43039511822808537942544375083, 10.93827757357396896192657810675, 12.17953235790664255762933585279, 12.776574956400157750201126803894, 13.4962245602750792444017942259, 14.17411708527755398915574643599, 14.61012987466438695188208340827, 15.51356271134500393435601971513, 16.531317475840807693723543651913, 16.92284825881041818606088932598, 18.25069011819090494357679186222, 19.225310458079459285129591027181, 19.64988950749912739189555236471, 20.66494723980063741569003312614

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