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Properties

Label 1-1441-1441.126-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.905 - 0.425i$
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  + (−0.0241 + 0.999i)2-s + (−0.443 − 0.896i)3-s + (−0.998 − 0.0483i)4-s + (0.715 − 0.698i)5-s + (0.906 − 0.421i)6-s + (0.120 − 0.992i)7-s + (0.0724 − 0.997i)8-s + (−0.607 + 0.794i)9-s + (0.681 + 0.732i)10-s + (0.399 + 0.916i)12-s + (0.120 − 0.992i)13-s + (0.989 + 0.144i)14-s + (−0.943 − 0.331i)15-s + (0.995 + 0.0965i)16-s + (−0.399 + 0.916i)17-s + (−0.779 − 0.626i)18-s + ⋯
L(s)  = 1  + (−0.0241 + 0.999i)2-s + (−0.443 − 0.896i)3-s + (−0.998 − 0.0483i)4-s + (0.715 − 0.698i)5-s + (0.906 − 0.421i)6-s + (0.120 − 0.992i)7-s + (0.0724 − 0.997i)8-s + (−0.607 + 0.794i)9-s + (0.681 + 0.732i)10-s + (0.399 + 0.916i)12-s + (0.120 − 0.992i)13-s + (0.989 + 0.144i)14-s + (−0.943 − 0.331i)15-s + (0.995 + 0.0965i)16-s + (−0.399 + 0.916i)17-s + (−0.779 − 0.626i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.905 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.905 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.905 - 0.425i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.905 - 0.425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2432381928 - 1.089270340i\)
\(L(\frac12)\) \(\approx\) \(0.2432381928 - 1.089270340i\)
\(L(1)\) \(\approx\) \(0.8613306336 - 0.1748012507i\)
\(L(1)\) \(\approx\) \(0.8613306336 - 0.1748012507i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.0241 + 0.999i)T \)
3 \( 1 + (-0.443 - 0.896i)T \)
5 \( 1 + (0.715 - 0.698i)T \)
7 \( 1 + (0.120 - 0.992i)T \)
13 \( 1 + (0.120 - 0.992i)T \)
17 \( 1 + (-0.399 + 0.916i)T \)
19 \( 1 + (0.861 + 0.506i)T \)
23 \( 1 + (-0.644 - 0.764i)T \)
29 \( 1 + (-0.568 + 0.822i)T \)
31 \( 1 + (0.527 + 0.849i)T \)
37 \( 1 + (0.354 - 0.935i)T \)
41 \( 1 + (0.215 - 0.976i)T \)
43 \( 1 + (0.715 - 0.698i)T \)
47 \( 1 + (0.607 - 0.794i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.399 - 0.916i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.715 - 0.698i)T \)
71 \( 1 + (-0.926 - 0.377i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.989 - 0.144i)T \)
83 \( 1 + (0.998 - 0.0483i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.981 + 0.192i)T \)
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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

−20.93165852432285217095765424328, −20.445130356578011281826045980571, −19.27086320964673850589633324961, −18.60557352517034699275396280913, −17.89262406617702614005316170131, −17.430587522510688436283907647413, −16.37369361581798473992240773435, −15.49803826113204034223952371764, −14.72464570138903930604704066801, −13.94335356775759296333835612846, −13.31630570012940178505731448055, −12.04813552260249952280521873759, −11.42825980326828850518659486756, −11.15243160885566806817064767934, −9.88067627577610757910516843862, −9.51487746787689119595957509370, −9.02182287591370821117381263428, −7.75322421277223318108493026093, −6.356184410123149952942192731604, −5.69886607381882550513850550192, −4.86528721567613619899745422747, −4.06070757710689056025527426822, −2.90742960502725350193889552614, −2.45141237814594426983064419381, −1.23617036961909981963490362005, 0.27224939270958835304217818033, 0.97760349822271245003796845105, 1.91217538016368879265055644217, 3.50747584052438914181207797276, 4.573275335035708104379180619710, 5.44195816890501625088919826257, 5.97717953295852944064107471652, 6.837469876331079119936000098828, 7.64212822731402553479615799720, 8.27199688873271888975447665900, 9.06974233280630037624459576428, 10.28630790419711865078993354330, 10.66365894672532573345696261068, 12.24424332079256043505990526037, 12.728496117947165244866242994250, 13.47980684989032115194061076841, 13.98818762073317664723650898817, 14.74261754686689794091257425115, 16.082272261959361074527412022758, 16.45838456893516035470357808678, 17.403960104363970864108230044010, 17.64914412628557104038770842765, 18.31416679813474794694379089452, 19.37516347796842153190110129223, 20.122428211015294021118203527468

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