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Properties

Label 1-33e2-1089.310-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.955 - 0.293i$
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  + (0.290 − 0.956i)2-s + (−0.830 − 0.556i)4-s + (0.988 − 0.151i)5-s + (−0.879 − 0.475i)7-s + (−0.774 + 0.633i)8-s + (0.142 − 0.989i)10-s + (−0.999 + 0.0380i)13-s + (−0.710 + 0.703i)14-s + (0.380 + 0.924i)16-s + (−0.198 − 0.980i)17-s + (−0.993 + 0.113i)19-s + (−0.905 − 0.424i)20-s + (0.723 + 0.690i)23-s + (0.953 − 0.299i)25-s + (−0.254 + 0.967i)26-s + ⋯
L(s)  = 1  + (0.290 − 0.956i)2-s + (−0.830 − 0.556i)4-s + (0.988 − 0.151i)5-s + (−0.879 − 0.475i)7-s + (−0.774 + 0.633i)8-s + (0.142 − 0.989i)10-s + (−0.999 + 0.0380i)13-s + (−0.710 + 0.703i)14-s + (0.380 + 0.924i)16-s + (−0.198 − 0.980i)17-s + (−0.993 + 0.113i)19-s + (−0.905 − 0.424i)20-s + (0.723 + 0.690i)23-s + (0.953 − 0.299i)25-s + (−0.254 + 0.967i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.955 - 0.293i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (310, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.955 - 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.282477548 - 0.1925186721i\)
\(L(\frac12)\) \(\approx\) \(1.282477548 - 0.1925186721i\)
\(L(1)\) \(\approx\) \(0.8714117610 - 0.5200937980i\)
\(L(1)\) \(\approx\) \(0.8714117610 - 0.5200937980i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.290 - 0.956i)T \)
5 \( 1 + (0.988 - 0.151i)T \)
7 \( 1 + (-0.879 - 0.475i)T \)
13 \( 1 + (-0.999 + 0.0380i)T \)
17 \( 1 + (-0.198 - 0.980i)T \)
19 \( 1 + (-0.993 + 0.113i)T \)
23 \( 1 + (0.723 + 0.690i)T \)
29 \( 1 + (0.217 + 0.976i)T \)
31 \( 1 + (-0.969 - 0.244i)T \)
37 \( 1 + (-0.985 + 0.170i)T \)
41 \( 1 + (-0.483 + 0.875i)T \)
43 \( 1 + (0.888 - 0.458i)T \)
47 \( 1 + (0.123 - 0.992i)T \)
53 \( 1 + (0.610 + 0.791i)T \)
59 \( 1 + (-0.999 - 0.0190i)T \)
61 \( 1 + (0.683 - 0.730i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (-0.870 - 0.491i)T \)
73 \( 1 + (0.564 + 0.825i)T \)
79 \( 1 + (-0.548 + 0.836i)T \)
83 \( 1 + (0.179 - 0.983i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.988 + 0.151i)T \)
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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

−21.51794891395745668606396951951, −20.85474777096310057721040190002, −19.39354382175016427483642071896, −18.94876110567420655514318623920, −17.91330632424826756894076340155, −17.199812756917499755571362287152, −16.760320028673986465726454247486, −15.743772922411247641868225771207, −14.95050688377950415647865734450, −14.419538487109964130603921534712, −13.42003282290454966811722956932, −12.755880991806995806971258743509, −12.29188616134138427382166409552, −10.713924420543528390954823795553, −9.91354671710029794273223491417, −9.12157910496784129548112299812, −8.50788840300282647344405781474, −7.24706347369005988383190757886, −6.53549086978818084654911293023, −5.90403124600924577950840068665, −5.1049594569959194488365467724, −4.07929869273035249764076988059, −2.94493535685463024250163797475, −2.082112496200834635111721648303, −0.301790985671378760845246232032, 0.77122790048156988636115642946, 1.9229487727591743294481040548, 2.737456530544764623640681193914, 3.614006448425070312987093383871, 4.77378880381665703804994594281, 5.40318654974758391717858677415, 6.4621996067151844917584127904, 7.2896331961848032257886480219, 8.86646454190455902601044692345, 9.34219958838946886435468662903, 10.15237648377279222758554304031, 10.68933293026795196585343943847, 11.78655142363037610821549901239, 12.72470787091534231178508884661, 13.12895700933447582457746104173, 13.97850207749760828795661064807, 14.59101436093455672076899273603, 15.667847484901043905814391909603, 16.89400084741222476821784321220, 17.256844473786248373149360689748, 18.3244829056473638824799651292, 18.95030570203969712836753598793, 19.91151800899039012281513912140, 20.29677169332804396318381711318, 21.34802846974141691836452119951

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