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Properties

Label 1-33e2-1089.20-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.0213 - 0.999i$
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  + (0.625 − 0.780i)2-s + (−0.217 − 0.976i)4-s + (0.969 − 0.244i)5-s + (−0.999 − 0.0190i)7-s + (−0.897 − 0.441i)8-s + (0.415 − 0.909i)10-s + (0.749 + 0.662i)13-s + (−0.640 + 0.768i)14-s + (−0.905 + 0.424i)16-s + (−0.610 + 0.791i)17-s + (−0.564 + 0.825i)19-s + (−0.449 − 0.893i)20-s + (0.327 + 0.945i)23-s + (0.879 − 0.475i)25-s + (0.985 − 0.170i)26-s + ⋯
L(s)  = 1  + (0.625 − 0.780i)2-s + (−0.217 − 0.976i)4-s + (0.969 − 0.244i)5-s + (−0.999 − 0.0190i)7-s + (−0.897 − 0.441i)8-s + (0.415 − 0.909i)10-s + (0.749 + 0.662i)13-s + (−0.640 + 0.768i)14-s + (−0.905 + 0.424i)16-s + (−0.610 + 0.791i)17-s + (−0.564 + 0.825i)19-s + (−0.449 − 0.893i)20-s + (0.327 + 0.945i)23-s + (0.879 − 0.475i)25-s + (0.985 − 0.170i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0213 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0213 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.0213 - 0.999i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.0213 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.177583034 - 2.224573056i\)
\(L(\frac12)\) \(\approx\) \(2.177583034 - 2.224573056i\)
\(L(1)\) \(\approx\) \(1.356081181 - 0.7605795321i\)
\(L(1)\) \(\approx\) \(1.356081181 - 0.7605795321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.625 - 0.780i)T \)
5 \( 1 + (0.969 - 0.244i)T \)
7 \( 1 + (-0.999 - 0.0190i)T \)
13 \( 1 + (0.749 + 0.662i)T \)
17 \( 1 + (-0.610 + 0.791i)T \)
19 \( 1 + (-0.564 + 0.825i)T \)
23 \( 1 + (0.327 + 0.945i)T \)
29 \( 1 + (0.851 - 0.524i)T \)
31 \( 1 + (0.00951 - 0.999i)T \)
37 \( 1 + (0.993 + 0.113i)T \)
41 \( 1 + (-0.161 - 0.986i)T \)
43 \( 1 + (0.928 - 0.371i)T \)
47 \( 1 + (0.710 - 0.703i)T \)
53 \( 1 + (-0.0855 - 0.996i)T \)
59 \( 1 + (0.935 - 0.353i)T \)
61 \( 1 + (0.988 - 0.151i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.941 + 0.336i)T \)
73 \( 1 + (-0.921 - 0.389i)T \)
79 \( 1 + (0.999 - 0.0380i)T \)
83 \( 1 + (0.290 + 0.956i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (-0.969 - 0.244i)T \)
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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

−21.76453791542272160857924310404, −20.82674331414088174341818708792, −20.107290386693052636423863633541, −18.9608602994297691942319564019, −17.95214467162285776822124928668, −17.66230264096321921405901974736, −16.504772542379934587303060434266, −16.04803257196498215853240883994, −15.17911154189657035620788465624, −14.35736177433690017715937541848, −13.49820640623166632626449816154, −13.07147249660568861776821979804, −12.358277490784958239496678345971, −11.087010322954513094358071301410, −10.28685513263323547057320881740, −9.1459564589336650497925654138, −8.732268849366719079080836543806, −7.409340456328684931587582901247, −6.51568629328414868368375690307, −6.19307536694041844901090270159, −5.15609832233311619310552055836, −4.28741045022522342302706705542, −2.951808024352388561919689786968, −2.64769761678762064318332050964, −0.79726839782272999991517339181, 0.667413799848636003401647, 1.77811710504395267584556652378, 2.494648730711423782025825542868, 3.683023801300189294123972925054, 4.30123300610211035504405448462, 5.63357697056290358860591321545, 6.09195806070458511808406658385, 6.8505862367844387398163152822, 8.52898732550814273712080019041, 9.2677971781636903617140777832, 9.98794002884967831978612954838, 10.638572202025990168229759048982, 11.61044194329188458688657796271, 12.53599444829595636872673427966, 13.249702181643014512010066570725, 13.61741651920920399973167973393, 14.56865836143634484767443608679, 15.46648906093359719122900743985, 16.32226183215651174056772218169, 17.22818020258775288287381200165, 18.07464270026857141952168024947, 19.09222112232837056312183369324, 19.36061741330826232968421354747, 20.601612868208294004840632750988, 20.948236074581450670722133852105

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