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Properties

Label 2-33e2-33.32-c1-0-32
Degree $2$
Conductor $1089$
Sign $0.542 + 0.840i$
Analytic cond. $8.69570$
Root an. cond. $2.94884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.43·2-s + 3.93·4-s − 3.79i·5-s + 0.367i·7-s + 4.71·8-s − 9.24i·10-s − 0.948i·13-s + 0.896i·14-s + 3.61·16-s + 3.43·17-s + 4.26i·19-s − 14.9i·20-s − 4.96i·23-s − 9.40·25-s − 2.31i·26-s + ⋯
L(s)  = 1  + 1.72·2-s + 1.96·4-s − 1.69i·5-s + 0.139i·7-s + 1.66·8-s − 2.92i·10-s − 0.263i·13-s + 0.239i·14-s + 0.904·16-s + 0.833·17-s + 0.977i·19-s − 3.34i·20-s − 1.03i·23-s − 1.88·25-s − 0.453i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.542 + 0.840i$
Analytic conductor: \(8.69570\)
Root analytic conductor: \(2.94884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1088, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :1/2),\ 0.542 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.388797987\)
\(L(\frac12)\) \(\approx\) \(4.388797987\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.43T + 2T^{2} \)
5 \( 1 + 3.79iT - 5T^{2} \)
7 \( 1 - 0.367iT - 7T^{2} \)
13 \( 1 + 0.948iT - 13T^{2} \)
17 \( 1 - 3.43T + 17T^{2} \)
19 \( 1 - 4.26iT - 19T^{2} \)
23 \( 1 + 4.96iT - 23T^{2} \)
29 \( 1 - 2.48T + 29T^{2} \)
31 \( 1 - 3.51T + 31T^{2} \)
37 \( 1 + 7.18T + 37T^{2} \)
41 \( 1 - 2.71T + 41T^{2} \)
43 \( 1 + 1.88iT - 43T^{2} \)
47 \( 1 + 0.0206iT - 47T^{2} \)
53 \( 1 - 5.54iT - 53T^{2} \)
59 \( 1 - 6.62iT - 59T^{2} \)
61 \( 1 - 9.75iT - 61T^{2} \)
67 \( 1 - 4.46T + 67T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 - 4.39iT - 73T^{2} \)
79 \( 1 - 10.9iT - 79T^{2} \)
83 \( 1 - 9.18T + 83T^{2} \)
89 \( 1 - 3.04iT - 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−9.847946430149090820865622077295, −8.732668637619730475367264981888, −8.084322754771491761221892897682, −6.95652258234472079571427249817, −5.78061328537636975770246521715, −5.42626413546100863333058960035, −4.49721766561130748577218125373, −3.87178822081599326389787737874, −2.60151406778585176700772035822, −1.23587956172885078500293519547, 2.11557891515205814980289649053, 3.11178710193257330203013000813, 3.61209653718164684571515139508, 4.77232178628069598076902810127, 5.70786665076761370824186779501, 6.56652750884672892229098167680, 7.03789357516585592719913887723, 7.87426001710740171584437790258, 9.425267799611143321362551770723, 10.40017080216950971344059417525

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