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Properties

Label 1-1441-1441.1008-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.118 + 0.992i$
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  + (−0.885 − 0.464i)2-s + (−0.0724 − 0.997i)3-s + (0.568 + 0.822i)4-s + (0.644 + 0.764i)5-s + (−0.399 + 0.916i)6-s + (0.861 + 0.506i)7-s + (−0.120 − 0.992i)8-s + (−0.989 + 0.144i)9-s + (−0.215 − 0.976i)10-s + (0.779 − 0.626i)12-s + (−0.995 + 0.0965i)13-s + (−0.527 − 0.849i)14-s + (0.715 − 0.698i)15-s + (−0.354 + 0.935i)16-s + (0.998 + 0.0483i)17-s + (0.943 + 0.331i)18-s + ⋯
L(s)  = 1  + (−0.885 − 0.464i)2-s + (−0.0724 − 0.997i)3-s + (0.568 + 0.822i)4-s + (0.644 + 0.764i)5-s + (−0.399 + 0.916i)6-s + (0.861 + 0.506i)7-s + (−0.120 − 0.992i)8-s + (−0.989 + 0.144i)9-s + (−0.215 − 0.976i)10-s + (0.779 − 0.626i)12-s + (−0.995 + 0.0965i)13-s + (−0.527 − 0.849i)14-s + (0.715 − 0.698i)15-s + (−0.354 + 0.935i)16-s + (0.998 + 0.0483i)17-s + (0.943 + 0.331i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.118 + 0.992i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1008, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.118 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4380092327 + 0.4932450110i\)
\(L(\frac12)\) \(\approx\) \(0.4380092327 + 0.4932450110i\)
\(L(1)\) \(\approx\) \(0.7290415432 - 0.1480561184i\)
\(L(1)\) \(\approx\) \(0.7290415432 - 0.1480561184i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.885 - 0.464i)T \)
3 \( 1 + (-0.0724 - 0.997i)T \)
5 \( 1 + (0.644 + 0.764i)T \)
7 \( 1 + (0.861 + 0.506i)T \)
13 \( 1 + (-0.995 + 0.0965i)T \)
17 \( 1 + (0.998 + 0.0483i)T \)
19 \( 1 + (0.262 - 0.964i)T \)
23 \( 1 + (0.485 + 0.873i)T \)
29 \( 1 + (-0.715 - 0.698i)T \)
31 \( 1 + (-0.681 - 0.732i)T \)
37 \( 1 + (-0.607 + 0.794i)T \)
41 \( 1 + (-0.836 - 0.548i)T \)
43 \( 1 + (0.0724 + 0.997i)T \)
47 \( 1 + (-0.443 - 0.896i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.354 + 0.935i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.0724 + 0.997i)T \)
71 \( 1 + (0.981 + 0.192i)T \)
73 \( 1 - T \)
79 \( 1 + (0.0724 + 0.997i)T \)
83 \( 1 + (0.943 - 0.331i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.215 - 0.976i)T \)
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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

−20.38232409002917913405629300109, −19.79437599589907121445747934499, −18.654301680120776239605993551368, −17.81700341450492632820024396886, −17.12716020533828992014878726995, −16.63575962258552159329471918233, −16.208978538424930585942418252553, −14.99240457514017991723823678332, −14.48017731152974734199933631694, −13.95834268049313763786613111845, −12.48019527779293549516591924356, −11.74202859919416464867725132241, −10.59528390871619151310695579273, −10.303158852974463664290130354719, −9.44029209725006480159389554135, −8.79560656634678322369622143977, −8.01131994151959139009760634959, −7.20796477460731656724041596442, −5.96254586049445309506860349207, −5.177107715550242544296984721462, −4.82450013279471866151527914661, −3.48928113743098609592982210223, −2.16469921823666718630954212731, −1.259923299924265860303155761111, −0.17331917103218996647027356168, 1.12780345301792287176243777377, 1.968711734854671955558341998184, 2.54909307051088350797951986321, 3.41909418316713989744696832245, 5.11525713299823382090970522491, 5.88396917769096785379314428028, 7.00458473071837682937384036225, 7.41679471776845658889115112098, 8.207548614845332717323274953534, 9.18430207327095470917361476088, 9.84433079570424007004553207498, 10.8666951801749466631553975843, 11.57207042692732620149712505565, 12.00851306500668862242600690628, 13.05748092780253807014817829061, 13.72717883691557609613128916607, 14.749555645058114796136326101858, 15.25867398863842597843420546081, 16.771110405508535487961297494317, 17.26590513973943347781932885797, 17.83372690747279588766554498219, 18.55638164025111269516198154324, 18.97486547609362403936473687240, 19.75273086215378939711438830111, 20.63450290083099813202276277574

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