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Properties

Label 1-1441-1441.1037-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.596 - 0.802i$
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  + (−0.120 − 0.992i)2-s + (−0.861 + 0.506i)3-s + (−0.970 + 0.239i)4-s + (0.995 + 0.0965i)5-s + (0.607 + 0.794i)6-s + (0.958 − 0.285i)7-s + (0.354 + 0.935i)8-s + (0.485 − 0.873i)9-s + (−0.0241 − 0.999i)10-s + (0.715 − 0.698i)12-s + (−0.943 − 0.331i)13-s + (−0.399 − 0.916i)14-s + (−0.906 + 0.421i)15-s + (0.885 − 0.464i)16-s + (0.168 + 0.985i)17-s + (−0.926 − 0.377i)18-s + ⋯
L(s)  = 1  + (−0.120 − 0.992i)2-s + (−0.861 + 0.506i)3-s + (−0.970 + 0.239i)4-s + (0.995 + 0.0965i)5-s + (0.607 + 0.794i)6-s + (0.958 − 0.285i)7-s + (0.354 + 0.935i)8-s + (0.485 − 0.873i)9-s + (−0.0241 − 0.999i)10-s + (0.715 − 0.698i)12-s + (−0.943 − 0.331i)13-s + (−0.399 − 0.916i)14-s + (−0.906 + 0.421i)15-s + (0.885 − 0.464i)16-s + (0.168 + 0.985i)17-s + (−0.926 − 0.377i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.596 - 0.802i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.596 - 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.713762013 - 0.8618613978i\)
\(L(\frac12)\) \(\approx\) \(1.713762013 - 0.8618613978i\)
\(L(1)\) \(\approx\) \(0.9294952258 - 0.3156633424i\)
\(L(1)\) \(\approx\) \(0.9294952258 - 0.3156633424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.120 - 0.992i)T \)
3 \( 1 + (-0.861 + 0.506i)T \)
5 \( 1 + (0.995 + 0.0965i)T \)
7 \( 1 + (0.958 - 0.285i)T \)
13 \( 1 + (-0.943 - 0.331i)T \)
17 \( 1 + (0.168 + 0.985i)T \)
19 \( 1 + (0.989 - 0.144i)T \)
23 \( 1 + (-0.836 + 0.548i)T \)
29 \( 1 + (0.906 + 0.421i)T \)
31 \( 1 + (0.262 - 0.964i)T \)
37 \( 1 + (0.0724 - 0.997i)T \)
41 \( 1 + (-0.443 - 0.896i)T \)
43 \( 1 + (-0.861 + 0.506i)T \)
47 \( 1 + (0.681 + 0.732i)T \)
53 \( 1 + T \)
59 \( 1 + (0.885 - 0.464i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.861 + 0.506i)T \)
71 \( 1 + (-0.779 + 0.626i)T \)
73 \( 1 - T \)
79 \( 1 + (0.861 - 0.506i)T \)
83 \( 1 + (-0.926 + 0.377i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.0241 + 0.999i)T \)
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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

−20.833144555641834681127561579074, −19.72548548651081845252203125011, −18.58500386998546958792237681684, −18.13434920075669554284706182883, −17.70922422854271778257803803261, −16.87105270988881564404148928292, −16.42988033776007720683705485360, −15.46300481652248574125114403880, −14.428688533461517316664307569765, −13.91185946642474115273492563460, −13.2988655946203255262734801100, −12.10305177451171719629305083410, −11.764654017006707683028536742641, −10.28101848125702407562920191083, −9.95851817260210347751970980691, −8.83701128306091434204233331309, −8.02455776931523667906008479143, −7.14507672835103095711183729039, −6.55202706701661290972467482988, −5.55646510003284451002643109049, −5.099240696647276680220181937982, −4.48192844328327570073277839595, −2.62912390612131783526225722172, −1.56133262746817834989414357286, −0.71702456145352079409778232003, 0.65255350768876775121111840249, 1.519161999511790226756383136149, 2.4040369264535192886360152366, 3.605454753388900660048520378821, 4.49160318276911168591189350355, 5.289966824274182785905726212459, 5.7796571337738260174299102699, 7.11646339719463113065598198756, 8.105888790237792241537278110308, 9.140633403275208945798281175928, 10.00024354393790634680585120373, 10.29160920443129862337788682316, 11.15379193036341505137273620630, 11.86406952518484708413666070794, 12.56037305180143311590885608062, 13.45599653167696377869247099328, 14.30415872745859292362858280589, 14.87106606113250048793957658962, 16.10189893950553681031674709835, 17.20389409388959048494588210392, 17.41893163101311058201289796245, 17.98444554881789863758235076027, 18.78843670105123837270461468264, 19.932787703554282061436585330367, 20.54631236211754505238060127548

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