Properties

Label 2-33e2-1.1-c3-0-76
Degree $2$
Conductor $1089$
Sign $-1$
Analytic cond. $64.2530$
Root an. cond. $8.01580$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.19·2-s + 19·4-s − 9·5-s + 24.2·7-s − 57.1·8-s + 46.7·10-s − 71.0·13-s − 126·14-s + 145·16-s + 88.3·17-s + 145.·19-s − 171·20-s − 90·23-s − 44·25-s + 369·26-s + 460.·28-s − 88.3·29-s − 188·31-s − 296.·32-s − 459·34-s − 218.·35-s + 133·37-s − 756·38-s + 514.·40-s + 36.3·41-s + 72.7·43-s + 467.·46-s + ⋯
L(s)  = 1  − 1.83·2-s + 2.37·4-s − 0.804·5-s + 1.30·7-s − 2.52·8-s + 1.47·10-s − 1.51·13-s − 2.40·14-s + 2.26·16-s + 1.26·17-s + 1.75·19-s − 1.91·20-s − 0.815·23-s − 0.351·25-s + 2.78·26-s + 3.10·28-s − 0.565·29-s − 1.08·31-s − 1.63·32-s − 2.31·34-s − 1.05·35-s + 0.590·37-s − 3.22·38-s + 2.03·40-s + 0.138·41-s + 0.257·43-s + 1.49·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(64.2530\)
Root analytic conductor: \(8.01580\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1089,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.19T + 8T^{2} \)
5 \( 1 + 9T + 125T^{2} \)
7 \( 1 - 24.2T + 343T^{2} \)
13 \( 1 + 71.0T + 2.19e3T^{2} \)
17 \( 1 - 88.3T + 4.91e3T^{2} \)
19 \( 1 - 145.T + 6.85e3T^{2} \)
23 \( 1 + 90T + 1.21e4T^{2} \)
29 \( 1 + 88.3T + 2.43e4T^{2} \)
31 \( 1 + 188T + 2.97e4T^{2} \)
37 \( 1 - 133T + 5.06e4T^{2} \)
41 \( 1 - 36.3T + 6.89e4T^{2} \)
43 \( 1 - 72.7T + 7.95e4T^{2} \)
47 \( 1 + 72T + 1.03e5T^{2} \)
53 \( 1 - 45T + 1.48e5T^{2} \)
59 \( 1 + 378T + 2.05e5T^{2} \)
61 \( 1 + 623.T + 2.26e5T^{2} \)
67 \( 1 + 386T + 3.00e5T^{2} \)
71 \( 1 - 198T + 3.57e5T^{2} \)
73 \( 1 + 76.2T + 3.89e5T^{2} \)
79 \( 1 - 152.T + 4.93e5T^{2} \)
83 \( 1 - 1.24e3T + 5.71e5T^{2} \)
89 \( 1 + 45T + 7.04e5T^{2} \)
97 \( 1 - 89T + 9.12e5T^{2} \)
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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

−9.168242950203991871831383684096, −8.007563073803696298667191800203, −7.59884238186714628150922456234, −7.39492203841073370192048441493, −5.82647576282541233308808360260, −4.84449317060603320844272764206, −3.37910566208498455015516457589, −2.13316797235437596218135857267, −1.15747541587775228975876494663, 0, 1.15747541587775228975876494663, 2.13316797235437596218135857267, 3.37910566208498455015516457589, 4.84449317060603320844272764206, 5.82647576282541233308808360260, 7.39492203841073370192048441493, 7.59884238186714628150922456234, 8.007563073803696298667191800203, 9.168242950203991871831383684096

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