Properties

Label 2-33e2-33.32-c1-0-18
Degree $2$
Conductor $1089$
Sign $0.384 - 0.923i$
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  + 2.35·2-s + 3.52·4-s + 2.24i·5-s + 4.05i·7-s + 3.58·8-s + 5.28i·10-s − 4.65i·13-s + 9.52i·14-s + 1.38·16-s − 0.0762·17-s + 2.39i·19-s + 7.92i·20-s + 3.22i·23-s − 0.0541·25-s − 10.9i·26-s + ⋯
L(s)  = 1  + 1.66·2-s + 1.76·4-s + 1.00i·5-s + 1.53i·7-s + 1.26·8-s + 1.67i·10-s − 1.29i·13-s + 2.54i·14-s + 0.345·16-s − 0.0185·17-s + 0.549i·19-s + 1.77i·20-s + 0.672i·23-s − 0.0108·25-s − 2.14i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.384 - 0.923i$
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: \((2,\ 1089,\ (\ :1/2),\ 0.384 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.048774761\)
\(L(\frac12)\) \(\approx\) \(4.048774761\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.35T + 2T^{2} \)
5 \( 1 - 2.24iT - 5T^{2} \)
7 \( 1 - 4.05iT - 7T^{2} \)
13 \( 1 + 4.65iT - 13T^{2} \)
17 \( 1 + 0.0762T + 17T^{2} \)
19 \( 1 - 2.39iT - 19T^{2} \)
23 \( 1 - 3.22iT - 23T^{2} \)
29 \( 1 + 1.83T + 29T^{2} \)
31 \( 1 - 1.67T + 31T^{2} \)
37 \( 1 - 7.26T + 37T^{2} \)
41 \( 1 - 8.44T + 41T^{2} \)
43 \( 1 + 4.28iT - 43T^{2} \)
47 \( 1 + 6.21iT - 47T^{2} \)
53 \( 1 + 1.22iT - 53T^{2} \)
59 \( 1 - 0.580iT - 59T^{2} \)
61 \( 1 + 3.78iT - 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 1.12iT - 71T^{2} \)
73 \( 1 + 13.3iT - 73T^{2} \)
79 \( 1 + 0.659iT - 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 6.58iT - 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−10.30089964826693525629368186186, −9.271691775450914837400949479495, −8.150242699916966106026594127750, −7.23751971161491581589094257469, −6.18756091684914369992323370290, −5.74920835291141333212232445829, −5.00379341037215657112067814429, −3.68757374116669778488601380295, −2.92506340093773297198305955221, −2.24001688825321628432367146531, 1.11118774541360845615008150573, 2.58868617198029418807404189560, 4.01544853398908682811097355121, 4.33323261740385186282611633805, 5.06235200070798610860542018950, 6.23699289491633186539199756247, 6.90589864741457303038765690168, 7.75372109146691084673897432356, 8.931106965621900910880300829857, 9.773315589352806634352252702929

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