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Properties

Label 1-1441-1441.1097-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.933 + 0.359i$
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  + (0.168 − 0.985i)2-s + (−0.0724 − 0.997i)3-s + (−0.943 − 0.331i)4-s + (0.644 + 0.764i)5-s + (−0.995 − 0.0965i)6-s + (0.748 − 0.663i)7-s + (−0.485 + 0.873i)8-s + (−0.989 + 0.144i)9-s + (0.861 − 0.506i)10-s + (−0.262 + 0.964i)12-s + (0.748 − 0.663i)13-s + (−0.527 − 0.849i)14-s + (0.715 − 0.698i)15-s + (0.779 + 0.626i)16-s + (0.262 + 0.964i)17-s + (−0.0241 + 0.999i)18-s + ⋯
L(s)  = 1  + (0.168 − 0.985i)2-s + (−0.0724 − 0.997i)3-s + (−0.943 − 0.331i)4-s + (0.644 + 0.764i)5-s + (−0.995 − 0.0965i)6-s + (0.748 − 0.663i)7-s + (−0.485 + 0.873i)8-s + (−0.989 + 0.144i)9-s + (0.861 − 0.506i)10-s + (−0.262 + 0.964i)12-s + (0.748 − 0.663i)13-s + (−0.527 − 0.849i)14-s + (0.715 − 0.698i)15-s + (0.779 + 0.626i)16-s + (0.262 + 0.964i)17-s + (−0.0241 + 0.999i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.933 + 0.359i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.933 + 0.359i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3666050568 - 1.970170177i\)
\(L(\frac12)\) \(\approx\) \(-0.3666050568 - 1.970170177i\)
\(L(1)\) \(\approx\) \(0.7739731538 - 0.8929827427i\)
\(L(1)\) \(\approx\) \(0.7739731538 - 0.8929827427i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.168 - 0.985i)T \)
3 \( 1 + (-0.0724 - 0.997i)T \)
5 \( 1 + (0.644 + 0.764i)T \)
7 \( 1 + (0.748 - 0.663i)T \)
13 \( 1 + (0.748 - 0.663i)T \)
17 \( 1 + (0.262 + 0.964i)T \)
19 \( 1 + (-0.836 - 0.548i)T \)
23 \( 1 + (0.981 - 0.192i)T \)
29 \( 1 + (-0.885 + 0.464i)T \)
31 \( 1 + (-0.681 - 0.732i)T \)
37 \( 1 + (0.568 + 0.822i)T \)
41 \( 1 + (0.998 - 0.0483i)T \)
43 \( 1 + (-0.644 - 0.764i)T \)
47 \( 1 + (-0.989 + 0.144i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.262 - 0.964i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.644 - 0.764i)T \)
71 \( 1 + (-0.906 + 0.421i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.527 - 0.849i)T \)
83 \( 1 + (0.943 - 0.331i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
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

−21.24535202971541359823827300253, −20.6846709570347335403020772276, −19.40237146328634121754871551710, −18.304019669264530322238663289318, −17.81000848410706625378854820575, −16.89515807789080612144712289056, −16.398667519409594202533796000332, −15.84449481827673316760094298686, −14.774357463978914079553493881929, −14.4981917632271959570643209123, −13.54216470995446481876652865943, −12.76115247064632023660196023712, −11.7658044507579327474987339, −10.97404347060408880908231452886, −9.7705288566121150593084823381, −9.160303571191248102788522250707, −8.67547127434720498567591824553, −7.89145509119119412571925528811, −6.59789535869675096379615319582, −5.69555815251295936551949104648, −5.23754244916798215360849965050, −4.49275493719824623470665593972, −3.69831050544637242346947482755, −2.3846277555888627425065103928, −1.05135564464881656402596227940, 0.392786246012995566403787537198, 1.41520955312802760836811453826, 1.99503896597770777065682142289, 2.990636922291637873481264085495, 3.802718015111731907814052585084, 5.039717540069728796076499990383, 5.861738118193211203508460839215, 6.639594077767149987166577128427, 7.69378321252289934892056987391, 8.41359184319190133823198613496, 9.332445974850071309298255800202, 10.52419699602811214204198976929, 10.92344020672619900008872473177, 11.44759214649427064297225672823, 12.72305443692114435376962601016, 13.14418350704655325578217359840, 13.746360603123196642224931280470, 14.69963579976379683251802742274, 14.95790107146596222986199386220, 16.96514557821309683119467812244, 17.294933920021809914392951735226, 18.10820439404293175916814612088, 18.61221220235713318905063515464, 19.28537794475192125139277776595, 20.16969082154906640893064824880

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