Properties

Label 2-33e2-33.32-c1-0-19
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  − 0.688·2-s − 1.52·4-s − 0.0401i·5-s + 0.246i·7-s + 2.42·8-s + 0.0276i·10-s − 2.30i·13-s − 0.169i·14-s + 1.38·16-s − 4.27·17-s + 6.20i·19-s + 0.0612i·20-s − 6.79i·23-s + 4.99·25-s + 1.58i·26-s + ⋯
L(s)  = 1  − 0.486·2-s − 0.763·4-s − 0.0179i·5-s + 0.0932i·7-s + 0.858·8-s + 0.00872i·10-s − 0.638i·13-s − 0.0454i·14-s + 0.345·16-s − 1.03·17-s + 1.42i·19-s + 0.0136i·20-s − 1.41i·23-s + 0.999·25-s + 0.310i·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

\(L(1)\) \(\approx\) \(0.7941804466\)
\(L(\frac12)\) \(\approx\) \(0.7941804466\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.688T + 2T^{2} \)
5 \( 1 + 0.0401iT - 5T^{2} \)
7 \( 1 - 0.246iT - 7T^{2} \)
13 \( 1 + 2.30iT - 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
19 \( 1 - 6.20iT - 19T^{2} \)
23 \( 1 + 6.79iT - 23T^{2} \)
29 \( 1 + 5.59T + 29T^{2} \)
31 \( 1 - 4.79T + 31T^{2} \)
37 \( 1 + 4.03T + 37T^{2} \)
41 \( 1 - 9.60T + 41T^{2} \)
43 \( 1 + 1.03iT - 43T^{2} \)
47 \( 1 + 11.1iT - 47T^{2} \)
53 \( 1 + 8.96iT - 53T^{2} \)
59 \( 1 + 2.78iT - 59T^{2} \)
61 \( 1 + 8.48iT - 61T^{2} \)
67 \( 1 - 7.94T + 67T^{2} \)
71 \( 1 + 3.32iT - 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 + 3.01iT - 79T^{2} \)
83 \( 1 + 5.29T + 83T^{2} \)
89 \( 1 + 8.54iT - 89T^{2} \)
97 \( 1 + 3.02T + 97T^{2} \)
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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

−9.709259270341238189946421873510, −8.740319946165314148553589741992, −8.341813322465757819307096544023, −7.40735584071940610329828522591, −6.37408112398852213898326472885, −5.35012247395157256996385435979, −4.48285572926991813837212151585, −3.53231076987696099132894774539, −2.07359307152150981892691724188, −0.51408885130907543948359990438, 1.11819915132474521782161659814, 2.61713481083898514510419594548, 4.03858853264800273785022775858, 4.69493841996800004765734577326, 5.71183822392350844525876943205, 6.93045106611899315665111024350, 7.53239586995472275178768725296, 8.661944624461309309707127280918, 9.151847031324253111812193096987, 9.749590656697379537345782249528

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