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Properties

Label 1-1441-1441.1033-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.986 - 0.165i$
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  + (0.836 − 0.548i)2-s + (−0.607 + 0.794i)3-s + (0.399 − 0.916i)4-s + (0.0241 − 0.999i)5-s + (−0.0724 + 0.997i)6-s + (−0.644 − 0.764i)7-s + (−0.168 − 0.985i)8-s + (−0.262 − 0.964i)9-s + (−0.527 − 0.849i)10-s + (0.485 + 0.873i)12-s + (0.527 − 0.849i)13-s + (−0.958 − 0.285i)14-s + (0.779 + 0.626i)15-s + (−0.681 − 0.732i)16-s + (0.981 + 0.192i)17-s + (−0.748 − 0.663i)18-s + ⋯
L(s)  = 1  + (0.836 − 0.548i)2-s + (−0.607 + 0.794i)3-s + (0.399 − 0.916i)4-s + (0.0241 − 0.999i)5-s + (−0.0724 + 0.997i)6-s + (−0.644 − 0.764i)7-s + (−0.168 − 0.985i)8-s + (−0.262 − 0.964i)9-s + (−0.527 − 0.849i)10-s + (0.485 + 0.873i)12-s + (0.527 − 0.849i)13-s + (−0.958 − 0.285i)14-s + (0.779 + 0.626i)15-s + (−0.681 − 0.732i)16-s + (0.981 + 0.192i)17-s + (−0.748 − 0.663i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.986 - 0.165i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.986 - 0.165i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1291263405 - 1.551084622i\)
\(L(\frac12)\) \(\approx\) \(0.1291263405 - 1.551084622i\)
\(L(1)\) \(\approx\) \(1.005083841 - 0.7401083280i\)
\(L(1)\) \(\approx\) \(1.005083841 - 0.7401083280i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.836 - 0.548i)T \)
3 \( 1 + (-0.607 + 0.794i)T \)
5 \( 1 + (0.0241 - 0.999i)T \)
7 \( 1 + (-0.644 - 0.764i)T \)
13 \( 1 + (0.527 - 0.849i)T \)
17 \( 1 + (0.981 + 0.192i)T \)
19 \( 1 + (0.120 - 0.992i)T \)
23 \( 1 + (-0.715 + 0.698i)T \)
29 \( 1 + (-0.262 + 0.964i)T \)
31 \( 1 + (0.443 - 0.896i)T \)
37 \( 1 + (-0.215 - 0.976i)T \)
41 \( 1 + (0.681 - 0.732i)T \)
43 \( 1 + (0.943 + 0.331i)T \)
47 \( 1 + (0.354 - 0.935i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.906 + 0.421i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.943 - 0.331i)T \)
71 \( 1 + (-0.885 + 0.464i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (0.995 - 0.0965i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.926 + 0.377i)T \)
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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

−21.42540609935944936247941635888, −20.53644109274392832055268428007, −19.18406782688510378827588337726, −18.820234488580868946009679540018, −18.117732501734478298959129071850, −17.257514424278572322302900841085, −16.3534967308787198529751437023, −15.91943527768568802748831886559, −14.91215165532148419458551754690, −14.03946863018061946429928726957, −13.7405642109342416557688314376, −12.498024540619801015418838871627, −12.17291297108109663516276975916, −11.41830250166567436591057646447, −10.56826129843864593355439489113, −9.48837478497059615996166365449, −8.18745818169639554438208849655, −7.639270962437374885405429711874, −6.59141388614475415403249594343, −6.215124648242707108961204688072, −5.65581728030714652628031485914, −4.465717812001572369550537855110, −3.3686599398868397819639533884, −2.62227899098754907129737885686, −1.69157194806498614186219702669, 0.49614493081052899118272257847, 1.27392711586258420054805519136, 2.8525921703067500209811845353, 3.82281837120428792934251657052, 4.18641041587132160227855424303, 5.429645129690609844716651960466, 5.602048095500019774192623312228, 6.69671182048909960275309530973, 7.80608133496560053256624734539, 9.1741256988612628505544559115, 9.67820963218594874550290844677, 10.52092139127500198310856601799, 11.08789272213429336035813730096, 12.08948337817323945444375533786, 12.6497516528740263966253339371, 13.37689426703145801733633872657, 14.13405243049311990727471882058, 15.19823706715644526065983638785, 15.89658465496785743744099785812, 16.32878419203128079896102164097, 17.2235655353825860845986897130, 17.97356861787568546843474117336, 19.23614969807164694330085313874, 19.98071604869467055148838131216, 20.47490355903911216689492184783

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