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Properties

Label 1-33e2-1089.31-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.838 - 0.544i$
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  + (0.749 + 0.662i)2-s + (0.123 + 0.992i)4-s + (−0.991 − 0.132i)5-s + (0.345 − 0.938i)7-s + (−0.564 + 0.825i)8-s + (−0.654 − 0.755i)10-s + (−0.683 − 0.730i)13-s + (0.879 − 0.475i)14-s + (−0.969 + 0.244i)16-s + (−0.362 + 0.931i)17-s + (0.774 − 0.633i)19-s + (0.00951 − 0.999i)20-s + (−0.786 + 0.618i)23-s + (0.964 + 0.263i)25-s + (−0.0285 − 0.999i)26-s + ⋯
L(s)  = 1  + (0.749 + 0.662i)2-s + (0.123 + 0.992i)4-s + (−0.991 − 0.132i)5-s + (0.345 − 0.938i)7-s + (−0.564 + 0.825i)8-s + (−0.654 − 0.755i)10-s + (−0.683 − 0.730i)13-s + (0.879 − 0.475i)14-s + (−0.969 + 0.244i)16-s + (−0.362 + 0.931i)17-s + (0.774 − 0.633i)19-s + (0.00951 − 0.999i)20-s + (−0.786 + 0.618i)23-s + (0.964 + 0.263i)25-s + (−0.0285 − 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.838 - 0.544i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.838 - 0.544i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329750130 - 0.3937856180i\)
\(L(\frac12)\) \(\approx\) \(1.329750130 - 0.3937856180i\)
\(L(1)\) \(\approx\) \(1.187244834 + 0.2102084600i\)
\(L(1)\) \(\approx\) \(1.187244834 + 0.2102084600i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.749 + 0.662i)T \)
5 \( 1 + (-0.991 - 0.132i)T \)
7 \( 1 + (0.345 - 0.938i)T \)
13 \( 1 + (-0.683 - 0.730i)T \)
17 \( 1 + (-0.362 + 0.931i)T \)
19 \( 1 + (0.774 - 0.633i)T \)
23 \( 1 + (-0.786 + 0.618i)T \)
29 \( 1 + (-0.710 - 0.703i)T \)
31 \( 1 + (0.820 - 0.572i)T \)
37 \( 1 + (0.516 - 0.856i)T \)
41 \( 1 + (-0.595 - 0.803i)T \)
43 \( 1 + (0.723 - 0.690i)T \)
47 \( 1 + (0.953 - 0.299i)T \)
53 \( 1 + (0.696 - 0.717i)T \)
59 \( 1 + (-0.398 - 0.917i)T \)
61 \( 1 + (-0.948 - 0.318i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (-0.998 + 0.0570i)T \)
73 \( 1 + (0.897 - 0.441i)T \)
79 \( 1 + (-0.761 + 0.647i)T \)
83 \( 1 + (0.999 + 0.0380i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.991 + 0.132i)T \)
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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

−21.68424851738267081958798679860, −20.62093018835667980335779484379, −20.13237498493719794770579133127, −19.251524151608305077251260400153, −18.57128525705170004046204727277, −18.07967498552597583560254318116, −16.51399545715254874972219627311, −15.87703261849942970067387722525, −15.07436698015340655435430341654, −14.463777445068023044654958352455, −13.72609889469734378621667550661, −12.52674361648159661618130937349, −11.912920294019199122136425686755, −11.59782731740337691978565660127, −10.60710841348669983053284415577, −9.60031142880351925738471822210, −8.82474322249726173633546837237, −7.72780799422002176360206025381, −6.815592403170474149331444246999, −5.820940552829641229658624136007, −4.821889220261113243970996706979, −4.291800938284839694801958481178, −3.09350393493744159781115889296, −2.456687824664447303420619853582, −1.24276700642988567268580482945, 0.46774127266053155232977780039, 2.212482169252861104684302692, 3.48592532656262791505329175195, 4.042961566457654022769629254266, 4.8585511386446856456865919533, 5.76196107605425076573640759452, 6.901378038927756881428832276630, 7.68056156559522480250151146304, 7.98118973653013345341424071157, 9.14111999390062069243614207750, 10.39850983360048519710358514005, 11.27995456851803623908674048155, 11.97994340971412043864484650314, 12.80855221796195529624332955434, 13.58026709794781708423721351855, 14.33595596593483255398330746018, 15.350544304094249704753049142218, 15.54818383966180763341297240410, 16.691743244256685316601108718262, 17.24768416396617508864588346048, 17.95376368655747836031238843503, 19.242130346485222085427713232359, 20.07109507540604208097236297855, 20.464244574891661212043692149170, 21.53229042596338559455252772173

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