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Properties

Label 1-33e2-1089.149-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.358 - 0.933i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (0.861 − 0.508i)2-s + (0.483 − 0.875i)4-s + (0.179 + 0.983i)5-s + (0.830 − 0.556i)7-s + (−0.0285 − 0.999i)8-s + (0.654 + 0.755i)10-s + (0.905 − 0.424i)13-s + (0.432 − 0.901i)14-s + (−0.532 − 0.846i)16-s + (−0.998 − 0.0570i)17-s + (0.254 − 0.967i)19-s + (0.948 + 0.318i)20-s + (0.786 − 0.618i)23-s + (−0.935 + 0.353i)25-s + (0.564 − 0.825i)26-s + ⋯
L(s)  = 1  + (0.861 − 0.508i)2-s + (0.483 − 0.875i)4-s + (0.179 + 0.983i)5-s + (0.830 − 0.556i)7-s + (−0.0285 − 0.999i)8-s + (0.654 + 0.755i)10-s + (0.905 − 0.424i)13-s + (0.432 − 0.901i)14-s + (−0.532 − 0.846i)16-s + (−0.998 − 0.0570i)17-s + (0.254 − 0.967i)19-s + (0.948 + 0.318i)20-s + (0.786 − 0.618i)23-s + (−0.935 + 0.353i)25-s + (0.564 − 0.825i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.358 - 0.933i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.358 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.445219101 - 1.680292888i\)
\(L(\frac12)\) \(\approx\) \(2.445219101 - 1.680292888i\)
\(L(1)\) \(\approx\) \(1.826734939 - 0.6872614788i\)
\(L(1)\) \(\approx\) \(1.826734939 - 0.6872614788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.861 - 0.508i)T \)
5 \( 1 + (0.179 + 0.983i)T \)
7 \( 1 + (0.830 - 0.556i)T \)
13 \( 1 + (0.905 - 0.424i)T \)
17 \( 1 + (-0.998 - 0.0570i)T \)
19 \( 1 + (0.254 - 0.967i)T \)
23 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (0.161 + 0.986i)T \)
31 \( 1 + (-0.290 - 0.956i)T \)
37 \( 1 + (-0.921 + 0.389i)T \)
41 \( 1 + (0.953 + 0.299i)T \)
43 \( 1 + (-0.723 + 0.690i)T \)
47 \( 1 + (0.595 - 0.803i)T \)
53 \( 1 + (0.466 + 0.884i)T \)
59 \( 1 + (0.217 - 0.976i)T \)
61 \( 1 + (-0.00951 + 0.999i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (0.362 + 0.931i)T \)
73 \( 1 + (0.985 + 0.170i)T \)
79 \( 1 + (-0.380 - 0.924i)T \)
83 \( 1 + (0.272 + 0.962i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.179 + 0.983i)T \)
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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

−21.321824424864241742283584381148, −21.117507946992565016183288295759, −20.40570878664159112412355437489, −19.42236462249226489131004773974, −18.23376699127247416648216929499, −17.498787112708563884276968742181, −16.85345516462401290734513628150, −15.84563508412103168310262372675, −15.53853058189167290942315594594, −14.41341447150898964773919594713, −13.76733571557349341720363424597, −13.02938911214939456489101898156, −12.20968665018249933235463598957, −11.56741054361879905251668423807, −10.74830910005570831826618916121, −9.22858071293842079540649034222, −8.6008056979047620223980195386, −7.949736896730973544204834214127, −6.85920070215246212482922205297, −5.83370488590360310251486567207, −5.27459672340936355741350506469, −4.41286595166701823351770248170, −3.64426722961684441606354863259, −2.2479265779468310671715711151, −1.45454805952831034386946245834, 0.98517076437920499429315870563, 2.10797759726771565201733237196, 2.95554174784463069470904023130, 3.8667496904506469263360876541, 4.738071045085021257812015261084, 5.6331733586198081797690632293, 6.69005959109286447998561110149, 7.14995263729086349546602392500, 8.41892739148132052929545575324, 9.52063468894897814366650485305, 10.64546533351273413274897254059, 10.957126843108255729760157045340, 11.53810007215052208087835022036, 12.78052018560522941450498207797, 13.57532209159716080641265406567, 14.02186495835371958708545487665, 15.039102886191848285470325732310, 15.34717703982920692995961089745, 16.4976362329857160241589074294, 17.68577939705322341258977871101, 18.220010002738287330478071152310, 19.03551352320920075290529186763, 20.03930226713708910200772333242, 20.49673932151336905056529427775, 21.423368948316075143125800649718

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