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Properties

Label 1-1441-1441.141-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.166 - 0.985i$
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  + (0.970 − 0.239i)2-s + (0.485 − 0.873i)3-s + (0.885 − 0.464i)4-s + (0.981 + 0.192i)5-s + (0.262 − 0.964i)6-s + (0.836 − 0.548i)7-s + (0.748 − 0.663i)8-s + (−0.527 − 0.849i)9-s + (0.998 − 0.0483i)10-s + (0.0241 − 0.999i)12-s + (0.779 + 0.626i)13-s + (0.681 − 0.732i)14-s + (0.644 − 0.764i)15-s + (0.568 − 0.822i)16-s + (0.943 − 0.331i)17-s + (−0.715 − 0.698i)18-s + ⋯
L(s)  = 1  + (0.970 − 0.239i)2-s + (0.485 − 0.873i)3-s + (0.885 − 0.464i)4-s + (0.981 + 0.192i)5-s + (0.262 − 0.964i)6-s + (0.836 − 0.548i)7-s + (0.748 − 0.663i)8-s + (−0.527 − 0.849i)9-s + (0.998 − 0.0483i)10-s + (0.0241 − 0.999i)12-s + (0.779 + 0.626i)13-s + (0.681 − 0.732i)14-s + (0.644 − 0.764i)15-s + (0.568 − 0.822i)16-s + (0.943 − 0.331i)17-s + (−0.715 − 0.698i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.166 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.166 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.166 - 0.985i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.166 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.325248339 - 6.301557652i\)
\(L(\frac12)\) \(\approx\) \(5.325248339 - 6.301557652i\)
\(L(1)\) \(\approx\) \(2.722282924 - 1.596734527i\)
\(L(1)\) \(\approx\) \(2.722282924 - 1.596734527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.970 - 0.239i)T \)
3 \( 1 + (0.485 - 0.873i)T \)
5 \( 1 + (0.981 + 0.192i)T \)
7 \( 1 + (0.836 - 0.548i)T \)
13 \( 1 + (0.779 + 0.626i)T \)
17 \( 1 + (0.943 - 0.331i)T \)
19 \( 1 + (-0.958 + 0.285i)T \)
23 \( 1 + (-0.399 + 0.916i)T \)
29 \( 1 + (-0.644 - 0.764i)T \)
31 \( 1 + (0.861 + 0.506i)T \)
37 \( 1 + (0.989 + 0.144i)T \)
41 \( 1 + (-0.607 + 0.794i)T \)
43 \( 1 + (0.485 - 0.873i)T \)
47 \( 1 + (0.0724 - 0.997i)T \)
53 \( 1 + T \)
59 \( 1 + (0.568 - 0.822i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.485 - 0.873i)T \)
71 \( 1 + (-0.215 + 0.976i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.485 + 0.873i)T \)
83 \( 1 + (-0.715 + 0.698i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.998 + 0.0483i)T \)
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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

−20.78596523947772313994618078169, −20.64203656539121488744109788254, −19.51286574524301729784937392119, −18.417530578205813142706913172125, −17.50190369125529997596986902643, −16.75365599783745943733937771769, −16.12896391907077947081157417507, −15.13225471085020662542543575939, −14.71686479442281380912895572906, −14.086047754968957202909900915188, −13.26027893613085021977185660875, −12.59552658339644111366828441156, −11.54370555016342287619828609926, −10.67317591747976837333410553931, −10.193158560002862075174099570591, −8.89387572327409625697169997930, −8.40544823551578648575401806680, −7.51868735508770080921693790955, −5.97659681394215511280043382052, −5.8084707136815377166779413126, −4.76847841360960084128611939402, −4.18368276859078799014085648702, −3.00506051955553165442865271195, −2.36110638546482373847462975523, −1.392841255396946437833124119, 0.996077596598094504247681482640, 1.67074319679520706760081154480, 2.33268419458810874355761450089, 3.401447907504051040072644904489, 4.22651015048933402939869911399, 5.39156986446756037790042570777, 6.08698802489576453095550313358, 6.81708137296617890333379637284, 7.6273714417305761295042718432, 8.49954524104319418503536305644, 9.64791343475897466150189155380, 10.414134049799943460254468750681, 11.40103578211775975151404916498, 11.92410549965607991293284980820, 13.02636342117302001796401956705, 13.519099589151573863217724583537, 14.118351525590783217190205569407, 14.587165970304111182036171555308, 15.44227787885063244961007047804, 16.740708852251497805503565557155, 17.241913554254328957482865801825, 18.344876447467970472791325453991, 18.77394410502261538905667769519, 19.7647604514307234057423053452, 20.50052389952715663104907701977

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