Properties

Label 2-33e2-11.10-c0-0-1
Degree $2$
Conductor $1089$
Sign $0.904 + 0.426i$
Analytic cond. $0.543481$
Root an. cond. $0.737212$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 1.41i·7-s + 1.41i·13-s + 16-s − 1.41i·19-s − 25-s − 1.41i·28-s + 1.41i·43-s − 1.00·49-s + 1.41i·52-s + 1.41i·61-s + 64-s + 1.41i·73-s − 1.41i·76-s − 1.41i·79-s + ⋯
L(s)  = 1  + 4-s − 1.41i·7-s + 1.41i·13-s + 16-s − 1.41i·19-s − 25-s − 1.41i·28-s + 1.41i·43-s − 1.00·49-s + 1.41i·52-s + 1.41i·61-s + 64-s + 1.41i·73-s − 1.41i·76-s − 1.41i·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ 0.904 + 0.426i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.286727564\)
\(L(\frac12)\) \(\approx\) \(1.286727564\)
\(L(1)\) 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 \( 1 - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−10.06721808329148398180358714857, −9.355268859818653488796930526917, −8.201978185270831678360105863139, −7.21026740810000805014704578668, −6.92379717397623357864914329454, −5.99374002314788415982377879248, −4.63826198419024545315704109965, −3.85803134120321147141647257358, −2.62409618595247610835911859261, −1.39050397970367248922835682035, 1.80435812220385207565973103636, 2.74495422624022727431092388158, 3.68341013037308675448901994764, 5.44367271914378711259131893485, 5.72178608052953435384779413066, 6.66798072466537257219409274280, 7.86999774936843300799674658882, 8.210923552930852750726430637685, 9.375810229740356501027215040802, 10.21096728128016961756596708127

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