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Properties

Label 1-1441-1441.1029-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.762 + 0.646i$
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  + (−0.485 − 0.873i)2-s + (−0.861 − 0.506i)3-s + (−0.527 + 0.849i)4-s + (0.995 − 0.0965i)5-s + (−0.0241 + 0.999i)6-s + (0.607 + 0.794i)7-s + (0.998 + 0.0483i)8-s + (0.485 + 0.873i)9-s + (−0.568 − 0.822i)10-s + (0.885 − 0.464i)12-s + (−0.0241 − 0.999i)13-s + (0.399 − 0.916i)14-s + (−0.906 − 0.421i)15-s + (−0.443 − 0.896i)16-s + (0.443 + 0.896i)17-s + (0.527 − 0.849i)18-s + ⋯
L(s)  = 1  + (−0.485 − 0.873i)2-s + (−0.861 − 0.506i)3-s + (−0.527 + 0.849i)4-s + (0.995 − 0.0965i)5-s + (−0.0241 + 0.999i)6-s + (0.607 + 0.794i)7-s + (0.998 + 0.0483i)8-s + (0.485 + 0.873i)9-s + (−0.568 − 0.822i)10-s + (0.885 − 0.464i)12-s + (−0.0241 − 0.999i)13-s + (0.399 − 0.916i)14-s + (−0.906 − 0.421i)15-s + (−0.443 − 0.896i)16-s + (0.443 + 0.896i)17-s + (0.527 − 0.849i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.762 + 0.646i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.762 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1525567064 - 0.4160606621i\)
\(L(\frac12)\) \(\approx\) \(-0.1525567064 - 0.4160606621i\)
\(L(1)\) \(\approx\) \(0.6067287103 - 0.3494112510i\)
\(L(1)\) \(\approx\) \(0.6067287103 - 0.3494112510i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.485 - 0.873i)T \)
3 \( 1 + (-0.861 - 0.506i)T \)
5 \( 1 + (0.995 - 0.0965i)T \)
7 \( 1 + (0.607 + 0.794i)T \)
13 \( 1 + (-0.0241 - 0.999i)T \)
17 \( 1 + (0.443 + 0.896i)T \)
19 \( 1 + (-0.715 - 0.698i)T \)
23 \( 1 + (-0.354 - 0.935i)T \)
29 \( 1 + (-0.981 - 0.192i)T \)
31 \( 1 + (-0.262 - 0.964i)T \)
37 \( 1 + (-0.527 + 0.849i)T \)
41 \( 1 + (-0.715 - 0.698i)T \)
43 \( 1 + (0.748 - 0.663i)T \)
47 \( 1 + (0.981 - 0.192i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.715 - 0.698i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.748 - 0.663i)T \)
71 \( 1 + (0.836 - 0.548i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.215 + 0.976i)T \)
83 \( 1 + (-0.926 - 0.377i)T \)
89 \( 1 + T \)
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

−21.076510704304950890042318890, −20.429499967381611101650793785696, −19.19383235838134774181328185970, −18.39899899182632917620010107254, −17.70737124656606702002615401061, −17.25995582442082075979013172266, −16.474717407363071142151907450875, −16.14840815987908344535943032775, −14.89686595339988723859867695496, −14.309956009043039723314653094317, −13.7238554704557355163655898264, −12.70585077398173482341567860232, −11.49241288114934184967620061890, −10.81320123594022973164855149853, −10.06032530385506587212758772987, −9.50703922440828333319785019647, −8.7158858260584856121839143789, −7.43580286329053062392968385363, −6.89777344744871966825135430236, −5.99887716080472272323469214860, −5.33422927030073440284498813180, −4.60733835973621034151620089649, −3.7152078133089400754877451215, −1.829828779043627923311020251772, −1.16176372726525592816989203670, 0.12165492012847428007730691111, 1.161318110538914747505301107301, 2.051946291647525379827579251796, 2.56270733474400428009101648407, 4.05225135226509614209351209153, 5.135924244773517472532894353901, 5.668986677725465767704154559936, 6.62235105886665012211302223761, 7.78287540198916011403857612108, 8.45379327943275457380722026998, 9.30164154255665309539648596463, 10.3877394104827853899341164135, 10.66976839746403180971445151931, 11.62279516093519916919668442286, 12.45457430581761556263275641199, 12.87617718339853209319410762079, 13.6206046095818267510923450284, 14.663505820782494603276847619219, 15.66229865664304089823698781909, 16.932787820594749610145378565743, 17.22304949493960955130378549133, 17.81305356314817075541315647207, 18.746182860507987233889528798180, 18.83838336121064134335524704359, 20.20662901539982065989297955967

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