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Properties

Label 1-33e2-1089.61-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.570 - 0.821i$
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  + (−0.548 − 0.836i)2-s + (−0.398 + 0.917i)4-s + (−0.532 − 0.846i)5-s + (−0.797 − 0.603i)7-s + (0.985 − 0.170i)8-s + (−0.415 + 0.909i)10-s + (−0.861 + 0.508i)13-s + (−0.0665 + 0.997i)14-s + (−0.683 − 0.730i)16-s + (−0.941 − 0.336i)17-s + (0.0285 + 0.999i)19-s + (0.988 − 0.151i)20-s + (−0.327 − 0.945i)23-s + (−0.432 + 0.901i)25-s + (0.897 + 0.441i)26-s + ⋯
L(s)  = 1  + (−0.548 − 0.836i)2-s + (−0.398 + 0.917i)4-s + (−0.532 − 0.846i)5-s + (−0.797 − 0.603i)7-s + (0.985 − 0.170i)8-s + (−0.415 + 0.909i)10-s + (−0.861 + 0.508i)13-s + (−0.0665 + 0.997i)14-s + (−0.683 − 0.730i)16-s + (−0.941 − 0.336i)17-s + (0.0285 + 0.999i)19-s + (0.988 − 0.151i)20-s + (−0.327 − 0.945i)23-s + (−0.432 + 0.901i)25-s + (0.897 + 0.441i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.570 - 0.821i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.570 - 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2403875279 - 0.1257333799i\)
\(L(\frac12)\) \(\approx\) \(0.2403875279 - 0.1257333799i\)
\(L(1)\) \(\approx\) \(0.4071924905 - 0.2398047614i\)
\(L(1)\) \(\approx\) \(0.4071924905 - 0.2398047614i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.548 - 0.836i)T \)
5 \( 1 + (-0.532 - 0.846i)T \)
7 \( 1 + (-0.797 - 0.603i)T \)
13 \( 1 + (-0.861 + 0.508i)T \)
17 \( 1 + (-0.941 - 0.336i)T \)
19 \( 1 + (0.0285 + 0.999i)T \)
23 \( 1 + (-0.327 - 0.945i)T \)
29 \( 1 + (-0.997 - 0.0760i)T \)
31 \( 1 + (-0.948 - 0.318i)T \)
37 \( 1 + (-0.736 - 0.676i)T \)
41 \( 1 + (0.710 + 0.703i)T \)
43 \( 1 + (-0.928 + 0.371i)T \)
47 \( 1 + (0.161 - 0.986i)T \)
53 \( 1 + (0.974 + 0.226i)T \)
59 \( 1 + (0.964 + 0.263i)T \)
61 \( 1 + (-0.449 - 0.893i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (-0.516 - 0.856i)T \)
79 \( 1 + (-0.272 + 0.962i)T \)
83 \( 1 + (-0.820 + 0.572i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (-0.532 + 0.846i)T \)
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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

−21.79138019256214731539912514802, −20.12959554059922248278920812232, −19.539477753130497750706423054780, −19.06473384346874451808108599695, −18.11764035943916289672998536211, −17.61940766777138934892510909018, −16.64153782971648040238114391261, −15.67339126382851514573664317804, −15.327778108936257941571526970905, −14.679215331296216746169868036066, −13.63067200887363600951693642229, −12.840674940518147202730160772923, −11.716504903760475294565331038103, −10.86184944932934553480434196618, −10.06184258846391337734375540828, −9.25532965291298734760152561819, −8.50830403623822906800080565414, −7.354143754106467743025393161883, −7.01315966136410277645199312158, −6.02499040774159704389300347934, −5.24005621196414540912711028176, −4.060681420575492322025526102707, −2.96861700495351751290044939784, −1.959462112294899740766660751356, −0.1979751394174416612274646523, 0.321111457328353430560744958932, 1.563467708845235523740096128796, 2.565889935846145979150785445515, 3.84012345907581938442216559818, 4.20642326008481559809696376976, 5.34333920662226118310010212510, 6.81387080863535441388836670375, 7.51645592963377491712171389501, 8.43048758852043795475565293513, 9.2556071235106454789884694053, 9.84631919649625015689170199391, 10.774452068078359253528290745757, 11.64396221318706678399128680813, 12.40779214535609402719320085623, 12.96350653304939487785927150481, 13.74717940960785452883418048448, 14.84862220016065473419245786040, 16.133902578202333782718741195300, 16.54621552204498894856198749783, 17.115968833805853570851976642197, 18.197517068602009339309873345080, 18.96942699377293845818432508669, 19.76124491678711403248256383589, 20.14814194932230712343218600207, 20.84778030394778809391366645462

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