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Properties

Label 1-1441-1441.1014-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.879 + 0.476i$
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  + (0.926 − 0.377i)2-s + (0.485 + 0.873i)3-s + (0.715 − 0.698i)4-s + (0.981 − 0.192i)5-s + (0.779 + 0.626i)6-s + (0.354 + 0.935i)7-s + (0.399 − 0.916i)8-s + (−0.527 + 0.849i)9-s + (0.836 − 0.548i)10-s + (0.958 + 0.285i)12-s + (0.354 + 0.935i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (0.0241 − 0.999i)16-s + (0.958 − 0.285i)17-s + (−0.168 + 0.985i)18-s + ⋯
L(s)  = 1  + (0.926 − 0.377i)2-s + (0.485 + 0.873i)3-s + (0.715 − 0.698i)4-s + (0.981 − 0.192i)5-s + (0.779 + 0.626i)6-s + (0.354 + 0.935i)7-s + (0.399 − 0.916i)8-s + (−0.527 + 0.849i)9-s + (0.836 − 0.548i)10-s + (0.958 + 0.285i)12-s + (0.354 + 0.935i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (0.0241 − 0.999i)16-s + (0.958 − 0.285i)17-s + (−0.168 + 0.985i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.879 + 0.476i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1014, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.879 + 0.476i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.183507933 + 1.061068403i\)
\(L(\frac12)\) \(\approx\) \(4.183507933 + 1.061068403i\)
\(L(1)\) \(\approx\) \(2.555069749 + 0.2881833052i\)
\(L(1)\) \(\approx\) \(2.555069749 + 0.2881833052i\)

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 \( 1 + (0.485 + 0.873i)T \)
5 \( 1 + (0.981 - 0.192i)T \)
7 \( 1 + (0.354 + 0.935i)T \)
13 \( 1 + (0.354 + 0.935i)T \)
17 \( 1 + (0.958 - 0.285i)T \)
19 \( 1 + (-0.607 - 0.794i)T \)
23 \( 1 + (-0.215 + 0.976i)T \)
29 \( 1 + (-0.970 + 0.239i)T \)
31 \( 1 + (0.861 - 0.506i)T \)
37 \( 1 + (-0.885 - 0.464i)T \)
41 \( 1 + (0.943 - 0.331i)T \)
43 \( 1 + (-0.981 + 0.192i)T \)
47 \( 1 + (0.527 - 0.849i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.958 - 0.285i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.981 - 0.192i)T \)
71 \( 1 + (-0.995 - 0.0965i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.681 + 0.732i)T \)
83 \( 1 + (0.715 + 0.698i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
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

−20.75116663499162183704206229404, −20.26448962412184120087263425317, −19.18084927052011107529151605621, −18.34614988613876177465558809545, −17.39543295751920314255725090363, −17.124586525506546474955571081536, −16.13203976912130738078627940996, −14.88172907969998404749940629723, −14.45215353053833226487153649595, −13.88817995967275305040546152106, −13.079050635544840730866235010800, −12.71005589180207865098660440582, −11.73523165581975177882189920517, −10.64557689372150831727287220729, −10.0816933326477236234143183277, −8.64196795219119771862568557099, −7.96837648950298782374377873765, −7.2905479216996060089595490359, −6.33720380439178436464568682410, −5.90648313604268845594785858890, −4.86151939673022798532899434149, −3.69793377560884555898763225963, −3.013512652127619825486911442236, −1.98997462254673836753838699993, −1.20075862405497155067452435892, 1.57156310711058015203154119709, 2.24224553884462204072460049182, 3.04447708850069593826312605755, 4.03596382825319362670634060642, 4.922974034714483107230758098291, 5.525331576447682187234715272527, 6.19818984368100022792572298013, 7.40372658065363299705732179254, 8.68493636614678105018608700649, 9.32164096080647018045559506327, 9.94829675279829419400715204526, 10.88263179756751790589655519964, 11.57320347539459196370406957553, 12.40074926486706057084877101657, 13.475571331159012815467066390435, 13.858000313176688350984874573205, 14.71515222995279608479900805585, 15.21549386709106176549816489312, 16.10065688093799219142110054544, 16.74404902727451945315264850801, 17.770189332938729245668790514262, 18.86458013852360656650981229410, 19.349242164191435055938384955811, 20.47362612588614035061285515460, 21.025370775443973561401793500541

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