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Properties

Label 2-33e2-11.8-c0-0-1
Degree $2$
Conductor $1089$
Sign $-0.255 + 0.966i$
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  + (0.309 − 0.951i)4-s + (−1.34 − 0.437i)7-s + (−0.831 − 1.14i)13-s + (−0.809 − 0.587i)16-s + (1.34 − 0.437i)19-s + (−0.309 − 0.951i)25-s + (−0.831 + 1.14i)28-s + 1.41i·43-s + (0.809 + 0.587i)49-s + (−1.34 + 0.437i)52-s + (0.831 − 1.14i)61-s + (−0.809 + 0.587i)64-s + (1.34 + 0.437i)73-s − 1.41i·76-s + (0.831 + 1.14i)79-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)4-s + (−1.34 − 0.437i)7-s + (−0.831 − 1.14i)13-s + (−0.809 − 0.587i)16-s + (1.34 − 0.437i)19-s + (−0.309 − 0.951i)25-s + (−0.831 + 1.14i)28-s + 1.41i·43-s + (0.809 + 0.587i)49-s + (−1.34 + 0.437i)52-s + (0.831 − 1.14i)61-s + (−0.809 + 0.587i)64-s + (1.34 + 0.437i)73-s − 1.41i·76-s + (0.831 + 1.14i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.255 + 0.966i$
Analytic conductor: \(0.543481\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1089,\ (\ :0),\ -0.255 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8326529209\)
\(L(\frac12)\) \(\approx\) \(0.8326529209\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.743845595838672347232538664352, −9.504501649496733303221856958298, −8.066446769714938404558133247546, −7.16579577214964534334286757147, −6.46425185041849582752605857257, −5.62759272033274842212929847937, −4.78907726706602976450175361737, −3.39370624573737144402028576322, −2.51332928671590422618556932016, −0.73652379186982093346515039048, 2.14535273795654806424150270438, 3.17705554271808749388478793727, 3.89275023384104356986805901376, 5.19585701014324314453978684123, 6.27179176729091800798712160692, 7.07020946031931932317995637474, 7.62198942973925509796427741921, 8.843536126443338871361876580556, 9.389863853636531788431732428665, 10.10883800679017843194707261204

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