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Properties

Label 2-33e2-11.6-c0-0-0
Degree $2$
Conductor $1089$
Sign $-0.0938 - 0.995i$
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  + (−0.809 + 0.587i)4-s + (0.831 + 1.14i)7-s + (−1.34 + 0.437i)13-s + (0.309 − 0.951i)16-s + (−0.831 + 1.14i)19-s + (0.809 + 0.587i)25-s + (−1.34 − 0.437i)28-s + 1.41i·43-s + (−0.309 + 0.951i)49-s + (0.831 − 1.14i)52-s + (1.34 + 0.437i)61-s + (0.309 + 0.951i)64-s + (−0.831 − 1.14i)73-s − 1.41i·76-s + (1.34 − 0.437i)79-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)4-s + (0.831 + 1.14i)7-s + (−1.34 + 0.437i)13-s + (0.309 − 0.951i)16-s + (−0.831 + 1.14i)19-s + (0.809 + 0.587i)25-s + (−1.34 − 0.437i)28-s + 1.41i·43-s + (−0.309 + 0.951i)49-s + (0.831 − 1.14i)52-s + (1.34 + 0.437i)61-s + (0.309 + 0.951i)64-s + (−0.831 − 1.14i)73-s − 1.41i·76-s + (1.34 − 0.437i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0938 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0938 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.0938 - 0.995i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (820, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ -0.0938 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7977144954\)
\(L(\frac12)\) \(\approx\) \(0.7977144954\)
\(L(1)\) 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 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
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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

−10.08378027266777955122824549369, −9.335315909755807050048303636413, −8.607123413207573461843242586894, −8.009256356972757416304581333066, −7.14052110300060959759244997026, −5.87022071246022123282800388836, −4.99048785404188417271629906794, −4.36092048291769080420979355836, −3.05361966177090638218079065431, −1.94848483513618685276119927009, 0.76198671179119282246770457432, 2.31917304404941302629406842304, 3.90734582682606064973369848354, 4.74484380664329132083583427079, 5.22346012892849327190364256642, 6.58894840832320962476003324715, 7.37456768357368292289237611418, 8.265057042609235963932222246621, 9.030564040896637976408327841118, 10.00287834441950506099861226647

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