Properties

Label 2-33e2-11.10-c2-0-14
Degree $2$
Conductor $1089$
Sign $-0.522 - 0.852i$
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  + 0.896i·2-s + 3.19·4-s − 8.66·5-s − 3.20i·7-s + 6.45i·8-s − 7.76i·10-s − 17.7i·13-s + 2.87·14-s + 7·16-s − 12.9i·17-s + 34.4i·19-s − 27.6·20-s + 10.7·23-s + 50.0·25-s + 15.9·26-s + ⋯
L(s)  = 1  + 0.448i·2-s + 0.799·4-s − 1.73·5-s − 0.458i·7-s + 0.806i·8-s − 0.776i·10-s − 1.36i·13-s + 0.205·14-s + 0.437·16-s − 0.762i·17-s + 1.81i·19-s − 1.38·20-s + 0.466·23-s + 2.00·25-s + 0.612·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.522 - 0.852i$
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: \((2,\ 1089,\ (\ :1),\ -0.522 - 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.072831786\)
\(L(\frac12)\) \(\approx\) \(1.072831786\)
\(L(2)\) 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 \( 1 - 0.896iT - 4T^{2} \)
5 \( 1 + 8.66T + 25T^{2} \)
7 \( 1 + 3.20iT - 49T^{2} \)
13 \( 1 + 17.7iT - 169T^{2} \)
17 \( 1 + 12.9iT - 289T^{2} \)
19 \( 1 - 34.4iT - 361T^{2} \)
23 \( 1 - 10.7T + 529T^{2} \)
29 \( 1 - 34.5iT - 841T^{2} \)
31 \( 1 + 8.53T + 961T^{2} \)
37 \( 1 + 35.1T + 1.36e3T^{2} \)
41 \( 1 - 40.3iT - 1.68e3T^{2} \)
43 \( 1 - 22.3iT - 1.84e3T^{2} \)
47 \( 1 - 9.89T + 2.20e3T^{2} \)
53 \( 1 + 23.1T + 2.80e3T^{2} \)
59 \( 1 + 96.4T + 3.48e3T^{2} \)
61 \( 1 - 61.0iT - 3.72e3T^{2} \)
67 \( 1 + 68.0T + 4.48e3T^{2} \)
71 \( 1 + 12.8T + 5.04e3T^{2} \)
73 \( 1 - 20.5iT - 5.32e3T^{2} \)
79 \( 1 - 109. iT - 6.24e3T^{2} \)
83 \( 1 - 30.3iT - 6.88e3T^{2} \)
89 \( 1 - 125.T + 7.92e3T^{2} \)
97 \( 1 + 46.8T + 9.40e3T^{2} \)
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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

−10.25623251620544084099841128152, −8.784969770311211623210647748822, −7.897474832119428165042946500237, −7.62240007942100699032699297896, −6.87161057858807661666244940796, −5.75866180061970508796326920092, −4.81172679530103090772952424543, −3.60506681733607766916962640806, −2.99763105381253383567706259814, −1.16147741928614257476793925862, 0.35524460394352115492525244248, 1.93794518816095063464213074576, 3.05919462113053597853912407766, 3.95768902470994993263876543102, 4.75190700734027067292923062734, 6.22402955668533862765081128017, 7.05721459099929860554711529549, 7.56523073201770988684389009854, 8.665112211630768626582685458601, 9.246329400310317683287876194666

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