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Properties

Label 2-33e2-11.10-c2-0-43
Degree $2$
Conductor $1089$
Sign $0.372 - 0.927i$
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  + 3.07i·2-s − 5.47·4-s − 4·5-s + 0.898i·7-s − 4.53i·8-s − 12.3i·10-s − 8.50i·13-s − 2.76·14-s − 7.94·16-s − 24.7i·17-s + 11.8i·19-s + 21.8·20-s + 7.23·23-s − 9·25-s + 26.1·26-s + ⋯
L(s)  = 1  + 1.53i·2-s − 1.36·4-s − 0.800·5-s + 0.128i·7-s − 0.566i·8-s − 1.23i·10-s − 0.654i·13-s − 0.197·14-s − 0.496·16-s − 1.45i·17-s + 0.624i·19-s + 1.09·20-s + 0.314·23-s − 0.359·25-s + 1.00·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.372 - 0.927i$
Analytic conductor: \(29.6731\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1),\ 0.372 - 0.927i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.243465345\)
\(L(\frac12)\) \(\approx\) \(1.243465345\)
\(L(2)\) 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 \( 1 - 3.07iT - 4T^{2} \)
5 \( 1 + 4T + 25T^{2} \)
7 \( 1 - 0.898iT - 49T^{2} \)
13 \( 1 + 8.50iT - 169T^{2} \)
17 \( 1 + 24.7iT - 289T^{2} \)
19 \( 1 - 11.8iT - 361T^{2} \)
23 \( 1 - 7.23T + 529T^{2} \)
29 \( 1 + 3.46iT - 841T^{2} \)
31 \( 1 - 33.1T + 961T^{2} \)
37 \( 1 - 40.2T + 1.36e3T^{2} \)
41 \( 1 + 1.30iT - 1.68e3T^{2} \)
43 \( 1 + 33.0iT - 1.84e3T^{2} \)
47 \( 1 + 22.7T + 2.20e3T^{2} \)
53 \( 1 - 78.5T + 2.80e3T^{2} \)
59 \( 1 + 31.2T + 3.48e3T^{2} \)
61 \( 1 - 28.0iT - 3.72e3T^{2} \)
67 \( 1 + 76.5T + 4.48e3T^{2} \)
71 \( 1 - 62.3T + 5.04e3T^{2} \)
73 \( 1 + 94.1iT - 5.32e3T^{2} \)
79 \( 1 - 65.9iT - 6.24e3T^{2} \)
83 \( 1 - 56.7iT - 6.88e3T^{2} \)
89 \( 1 - 62.2T + 7.92e3T^{2} \)
97 \( 1 - 72.4T + 9.40e3T^{2} \)
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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

−9.501596950450486762150612086219, −8.703751413638769215744651275086, −7.86461139792563280408664680456, −7.47225918144445960541982289732, −6.56901894183215736128440586844, −5.67311910389124046665553135231, −4.90423325069693016767500345821, −3.96483852507239753342913204407, −2.66452834322818774315286715840, −0.53237918866483149563176950493, 0.863041628387778892829239717971, 2.04959420830303774468371384474, 3.14531416173286865224890021001, 4.06411913434107265078866979670, 4.59741990788380711584796739462, 6.09103962341931874258136036438, 7.10303084447568804025431362874, 8.140039777066700742460387297728, 8.893361858892043576505057759912, 9.759960624330986994180541216267

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