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Properties

Label 1-29e2-841.639-r0-0-0
Degree $1$
Conductor $841$
Sign $0.106 + 0.994i$
Analytic cond. $3.90558$
Root an. cond. $3.90558$
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  + (0.0541 − 0.998i)2-s + (−0.725 + 0.687i)3-s + (−0.994 − 0.108i)4-s + (0.796 + 0.605i)5-s + (0.647 + 0.762i)6-s + (0.0541 + 0.998i)7-s + (−0.161 + 0.986i)8-s + (0.0541 − 0.998i)9-s + (0.647 − 0.762i)10-s + (−0.947 + 0.319i)11-s + (0.796 − 0.605i)12-s + (0.267 + 0.963i)13-s + 14-s + (−0.994 + 0.108i)15-s + (0.976 + 0.214i)16-s + (0.267 − 0.963i)17-s + ⋯
L(s)  = 1  + (0.0541 − 0.998i)2-s + (−0.725 + 0.687i)3-s + (−0.994 − 0.108i)4-s + (0.796 + 0.605i)5-s + (0.647 + 0.762i)6-s + (0.0541 + 0.998i)7-s + (−0.161 + 0.986i)8-s + (0.0541 − 0.998i)9-s + (0.647 − 0.762i)10-s + (−0.947 + 0.319i)11-s + (0.796 − 0.605i)12-s + (0.267 + 0.963i)13-s + 14-s + (−0.994 + 0.108i)15-s + (0.976 + 0.214i)16-s + (0.267 − 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.106 + 0.994i$
Analytic conductor: \(3.90558\)
Root analytic conductor: \(3.90558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 841,\ (0:\ ),\ 0.106 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7101342592 + 0.6382878477i\)
\(L(\frac12)\) \(\approx\) \(0.7101342592 + 0.6382878477i\)
\(L(1)\) \(\approx\) \(0.8452158957 + 0.08904659891i\)
\(L(1)\) \(\approx\) \(0.8452158957 + 0.08904659891i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (0.0541 - 0.998i)T \)
3 \( 1 + (-0.725 + 0.687i)T \)
5 \( 1 + (0.796 + 0.605i)T \)
7 \( 1 + (0.0541 + 0.998i)T \)
11 \( 1 + (-0.947 + 0.319i)T \)
13 \( 1 + (0.267 + 0.963i)T \)
17 \( 1 + (0.267 - 0.963i)T \)
19 \( 1 + (0.647 + 0.762i)T \)
23 \( 1 + (0.907 - 0.419i)T \)
31 \( 1 + (-0.725 + 0.687i)T \)
37 \( 1 + (0.907 - 0.419i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.994 - 0.108i)T \)
47 \( 1 + (-0.856 - 0.515i)T \)
53 \( 1 + (0.267 + 0.963i)T \)
59 \( 1 + (-0.994 + 0.108i)T \)
61 \( 1 + (0.267 + 0.963i)T \)
67 \( 1 + (0.976 + 0.214i)T \)
71 \( 1 + (-0.947 - 0.319i)T \)
73 \( 1 + (-0.947 + 0.319i)T \)
79 \( 1 + (-0.370 + 0.928i)T \)
83 \( 1 + (-0.856 - 0.515i)T \)
89 \( 1 + (-0.725 - 0.687i)T \)
97 \( 1 + (0.976 + 0.214i)T \)
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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

−22.1151432671085433144166143310, −21.39846960821118625073948779146, −20.37034257258718766545686070399, −19.3644109162890408690930209537, −18.308247665082247973498753872114, −17.75318207155018905399705340608, −17.137082808968021158810300519025, −16.52841901687864356784140282910, −15.766751440150059701727403532892, −14.63323023579381824953040370577, −13.580308409129268167805318336358, −13.084671434198714786131649769832, −12.7923228529160621506191360865, −11.211569050019859306613392608003, −10.385873331105188869599992409508, −9.56024981045130124169859749839, −8.2537211058679617575100286784, −7.759095970699447316625014895551, −6.81871727931090154868769157416, −5.89031883462063002455563631145, −5.32157062321656032713637451880, −4.535879864625369100988370595037, −3.09676133712600104764360729885, −1.41028799242136912956967870178, −0.517607052570881599347826208124, 1.416869819104094704706632119767, 2.54698978646758699032346245883, 3.2526126490938284998796495379, 4.57078098880858930635887498362, 5.35265237172837574966659460977, 5.9356218277953997334665123879, 7.19901047173561095102601433362, 8.73565534213443135311710525288, 9.46915873287554181605492096208, 10.02892453175921993119132181345, 10.92947105701891303592256405633, 11.54358044048891012601070035569, 12.35996806743581622514125677848, 13.204089007038411679710107160527, 14.29565465873197257703756154555, 14.85362756176546989317353209970, 15.96976494151630927440243687317, 16.78960037458274797450681657327, 17.920824068959922038159020677631, 18.35382081553193041825427580548, 18.78039050561582738162011355890, 20.22510729912188594641336962121, 21.17288135705585294675846257602, 21.30892052957709505462604313034, 22.11527582397744725805795173217

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