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Properties

Label 2-33e2-3.2-c2-0-14
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  + 2.99i·2-s − 4.96·4-s + 1.61i·5-s + 3.32·7-s − 2.90i·8-s − 4.82·10-s − 7.56·13-s + 9.97i·14-s − 11.1·16-s + 28.3i·17-s + 26.0·19-s − 8.00i·20-s + 19.5i·23-s + 22.4·25-s − 22.6i·26-s + ⋯
L(s)  = 1  + 1.49i·2-s − 1.24·4-s + 0.322i·5-s + 0.475·7-s − 0.362i·8-s − 0.482·10-s − 0.581·13-s + 0.712i·14-s − 0.698·16-s + 1.66i·17-s + 1.37·19-s − 0.400i·20-s + 0.850i·23-s + 0.896·25-s − 0.871i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.298093668\)
\(L(\frac12)\) \(\approx\) \(1.298093668\)
\(L(2)\) 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 \( 1 - 2.99iT - 4T^{2} \)
5 \( 1 - 1.61iT - 25T^{2} \)
7 \( 1 - 3.32T + 49T^{2} \)
13 \( 1 + 7.56T + 169T^{2} \)
17 \( 1 - 28.3iT - 289T^{2} \)
19 \( 1 - 26.0T + 361T^{2} \)
23 \( 1 - 19.5iT - 529T^{2} \)
29 \( 1 + 2.15iT - 841T^{2} \)
31 \( 1 + 9.22T + 961T^{2} \)
37 \( 1 + 67.5T + 1.36e3T^{2} \)
41 \( 1 + 29.0iT - 1.68e3T^{2} \)
43 \( 1 + 0.719T + 1.84e3T^{2} \)
47 \( 1 - 24.5iT - 2.20e3T^{2} \)
53 \( 1 - 5.79iT - 2.80e3T^{2} \)
59 \( 1 - 73.9iT - 3.48e3T^{2} \)
61 \( 1 + 72.1T + 3.72e3T^{2} \)
67 \( 1 + 79.8T + 4.48e3T^{2} \)
71 \( 1 - 107. iT - 5.04e3T^{2} \)
73 \( 1 - 90.3T + 5.32e3T^{2} \)
79 \( 1 + 114.T + 6.24e3T^{2} \)
83 \( 1 + 125. iT - 6.88e3T^{2} \)
89 \( 1 + 83.3iT - 7.92e3T^{2} \)
97 \( 1 + 78.0T + 9.40e3T^{2} \)
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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

−10.06455228272346260141078975195, −9.032440269747470317648088518136, −8.378548813491105811695598100255, −7.48087950305161362304975448834, −7.07153516726530931499664373327, −5.98521781733796648684059230160, −5.37337432768784748731678204635, −4.45813941339647462197050712129, −3.22874406086991300521783142808, −1.66039979811011384558957396947, 0.40029535427700379262812180900, 1.49720553241703831401654629329, 2.65820704673716175095821326754, 3.42072700327594605586647591850, 4.79434389637025157011299822795, 5.08228195439237173457868781259, 6.73990497929491443760378903538, 7.55288723223132638873717519526, 8.690401735366234755249607666550, 9.400002835250040837581214708107

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