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Properties

Label 1-33e2-1089.349-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.336 + 0.941i$
Analytic cond. $117.029$
Root an. cond. $117.029$
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  + (0.0665 + 0.997i)2-s + (−0.991 + 0.132i)4-s + (−0.217 − 0.976i)5-s + (−0.449 − 0.893i)7-s + (−0.198 − 0.980i)8-s + (0.959 − 0.281i)10-s + (0.432 + 0.901i)13-s + (0.861 − 0.508i)14-s + (0.964 − 0.263i)16-s + (0.921 + 0.389i)17-s + (−0.974 − 0.226i)19-s + (0.345 + 0.938i)20-s + (−0.888 + 0.458i)23-s + (−0.905 + 0.424i)25-s + (−0.870 + 0.491i)26-s + ⋯
L(s)  = 1  + (0.0665 + 0.997i)2-s + (−0.991 + 0.132i)4-s + (−0.217 − 0.976i)5-s + (−0.449 − 0.893i)7-s + (−0.198 − 0.980i)8-s + (0.959 − 0.281i)10-s + (0.432 + 0.901i)13-s + (0.861 − 0.508i)14-s + (0.964 − 0.263i)16-s + (0.921 + 0.389i)17-s + (−0.974 − 0.226i)19-s + (0.345 + 0.938i)20-s + (−0.888 + 0.458i)23-s + (−0.905 + 0.424i)25-s + (−0.870 + 0.491i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.336 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9221424889 + 0.6496636810i\)
\(L(\frac12)\) \(\approx\) \(0.9221424889 + 0.6496636810i\)
\(L(1)\) \(\approx\) \(0.7974474245 + 0.2341544329i\)
\(L(1)\) \(\approx\) \(0.7974474245 + 0.2341544329i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0665 + 0.997i)T \)
5 \( 1 + (-0.217 - 0.976i)T \)
7 \( 1 + (-0.449 - 0.893i)T \)
13 \( 1 + (0.432 + 0.901i)T \)
17 \( 1 + (0.921 + 0.389i)T \)
19 \( 1 + (-0.974 - 0.226i)T \)
23 \( 1 + (-0.888 + 0.458i)T \)
29 \( 1 + (-0.820 + 0.572i)T \)
31 \( 1 + (-0.851 - 0.524i)T \)
37 \( 1 + (0.941 + 0.336i)T \)
41 \( 1 + (-0.999 - 0.0380i)T \)
43 \( 1 + (0.995 - 0.0950i)T \)
47 \( 1 + (0.272 - 0.962i)T \)
53 \( 1 + (-0.254 - 0.967i)T \)
59 \( 1 + (-0.532 - 0.846i)T \)
61 \( 1 + (0.830 + 0.556i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (0.362 + 0.931i)T \)
79 \( 1 + (0.595 + 0.803i)T \)
83 \( 1 + (-0.161 + 0.986i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (-0.217 + 0.976i)T \)
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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

−21.12304918467035795744966435161, −20.31435649433583586790836013256, −19.45628820949207118221346609697, −18.735683602781851792769005106, −18.3690664059224243867080829977, −17.571481406576332957853110768223, −16.41306025331840836918568474428, −15.39664797387387677698092839448, −14.72636185314361681403425476173, −14.019258327425205545621838159806, −12.94826827007576248636073248987, −12.37565267087307483097992100678, −11.55808698151548469213182539407, −10.74548158361573726967140347755, −10.11248275442874443566837058808, −9.286920566521241947187309405261, −8.32301132834943423660096956362, −7.52945253180740122623374465062, −6.08172112517128883579308867643, −5.67865309412233602095125880727, −4.29832061705391775090019155206, −3.38079766454503817026496606247, −2.747242368188902920994814770092, −1.86724935667816423818584040842, −0.37963619365573241354742972663, 0.60513561457296994552666964707, 1.70488638910255862072358506430, 3.73207277144604043177541947466, 3.99522584043939305391139940909, 5.06695656776071109732539073651, 5.95179559974949697275346593267, 6.79613930342567500019394923919, 7.689410200039743295438494403007, 8.37025702840963267027959355499, 9.28392412795216938493505915882, 9.88876238752184900841791553544, 11.04994242474423551247211680401, 12.21099578958866390383993332880, 12.95088547749794052115119102560, 13.559209091400901540929076038744, 14.35358459373519698188186964884, 15.24973054766278615255750523593, 16.15898438609551763959474350930, 16.7441485269383262845605230788, 17.02423199820198814767325368003, 18.16841718687585207446994804954, 19.0408095682125613103544336950, 19.74758727077377212811619273794, 20.670568056362129043849676963790, 21.46426122219458169739905980504

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