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Properties

Label 2-33e2-3.2-c2-0-29
Degree $2$
Conductor $1089$
Sign $0.577 - 0.816i$
Analytic cond. $29.6731$
Root an. cond. $5.44730$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.356i·2-s + 3.87·4-s + 3.44i·5-s + 3.96·7-s − 2.80i·8-s + 1.22·10-s − 4.75·13-s − 1.41i·14-s + 14.4·16-s + 29.1i·17-s − 14.8·19-s + 13.3i·20-s + 22.9i·23-s + 13.1·25-s + 1.69i·26-s + ⋯
L(s)  = 1  − 0.178i·2-s + 0.968·4-s + 0.689i·5-s + 0.566·7-s − 0.350i·8-s + 0.122·10-s − 0.365·13-s − 0.101i·14-s + 0.905·16-s + 1.71i·17-s − 0.782·19-s + 0.667i·20-s + 0.998i·23-s + 0.525·25-s + 0.0651i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(29.6731\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.518819284\)
\(L(\frac12)\) \(\approx\) \(2.518819284\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + 0.356iT - 4T^{2} \)
5 \( 1 - 3.44iT - 25T^{2} \)
7 \( 1 - 3.96T + 49T^{2} \)
13 \( 1 + 4.75T + 169T^{2} \)
17 \( 1 - 29.1iT - 289T^{2} \)
19 \( 1 + 14.8T + 361T^{2} \)
23 \( 1 - 22.9iT - 529T^{2} \)
29 \( 1 - 4.63iT - 841T^{2} \)
31 \( 1 - 27.4T + 961T^{2} \)
37 \( 1 - 36.8T + 1.36e3T^{2} \)
41 \( 1 + 68.3iT - 1.68e3T^{2} \)
43 \( 1 + 26.8T + 1.84e3T^{2} \)
47 \( 1 - 42.9iT - 2.20e3T^{2} \)
53 \( 1 - 78.2iT - 2.80e3T^{2} \)
59 \( 1 - 92.7iT - 3.48e3T^{2} \)
61 \( 1 - 40.5T + 3.72e3T^{2} \)
67 \( 1 + 54.3T + 4.48e3T^{2} \)
71 \( 1 + 91.0iT - 5.04e3T^{2} \)
73 \( 1 + 109.T + 5.32e3T^{2} \)
79 \( 1 - 143.T + 6.24e3T^{2} \)
83 \( 1 - 44.0iT - 6.88e3T^{2} \)
89 \( 1 + 64.7iT - 7.92e3T^{2} \)
97 \( 1 - 16.8T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.17812208511940046902727337154, −8.939387202305702257306743872091, −7.970742629480629918376967787161, −7.33361170277100272952121748674, −6.40617413544892622218620089697, −5.80334291876622830296342908385, −4.44914772701892423288343979914, −3.39226580741874765350781342214, −2.38213585923246679344630475113, −1.42658275988193732100407971607, 0.76401770791476745414658371733, 2.09201628835237726104402598480, 3.01124383649700552699108117391, 4.60717939824864066822573971647, 5.09990130852736522225462523450, 6.32809750336805255085471587610, 6.97146993645658815692035097986, 7.976410793738103177510937030235, 8.499331705634766999045484248163, 9.594956438456352994449364324305

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