Properties

Label 2-33e2-1.1-c3-0-63
Degree $2$
Conductor $1089$
Sign $-1$
Analytic cond. $64.2530$
Root an. cond. $8.01580$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s − 18·5-s + 64·16-s + 144·20-s + 108·23-s + 199·25-s + 340·31-s − 434·37-s + 36·47-s − 343·49-s + 738·53-s + 720·59-s − 512·64-s − 416·67-s − 612·71-s − 1.15e3·80-s − 1.67e3·89-s − 864·92-s − 34·97-s − 1.59e3·100-s + 1.17e3·103-s − 2.14e3·113-s − 1.94e3·115-s + ⋯
L(s)  = 1  − 4-s − 1.60·5-s + 16-s + 1.60·20-s + 0.979·23-s + 1.59·25-s + 1.96·31-s − 1.92·37-s + 0.111·47-s − 49-s + 1.91·53-s + 1.58·59-s − 64-s − 0.758·67-s − 1.02·71-s − 1.60·80-s − 1.99·89-s − 0.979·92-s − 0.0355·97-s − 1.59·100-s + 1.12·103-s − 1.78·113-s − 1.57·115-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: yes
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 \( 1 + p^{3} T^{2} \)
5 \( 1 + 18 T + p^{3} T^{2} \)
7 \( 1 + p^{3} T^{2} \)
13 \( 1 + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 + p^{3} T^{2} \)
23 \( 1 - 108 T + p^{3} T^{2} \)
29 \( 1 + p^{3} T^{2} \)
31 \( 1 - 340 T + p^{3} T^{2} \)
37 \( 1 + 434 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 + p^{3} T^{2} \)
47 \( 1 - 36 T + p^{3} T^{2} \)
53 \( 1 - 738 T + p^{3} T^{2} \)
59 \( 1 - 720 T + p^{3} T^{2} \)
61 \( 1 + p^{3} T^{2} \)
67 \( 1 + 416 T + p^{3} T^{2} \)
71 \( 1 + 612 T + p^{3} T^{2} \)
73 \( 1 + p^{3} T^{2} \)
79 \( 1 + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + 1674 T + p^{3} T^{2} \)
97 \( 1 + 34 T + p^{3} T^{2} \)
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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

−8.736785677830695573996941291180, −8.446377138974293221095131063340, −7.53715153686217732863227816235, −6.75574963478171206685547455048, −5.37322498552637436041865688150, −4.56837298844257880324597562311, −3.85429552969987359634357864643, −2.98709261059238423344664646162, −1.00767604415401823796887979437, 0, 1.00767604415401823796887979437, 2.98709261059238423344664646162, 3.85429552969987359634357864643, 4.56837298844257880324597562311, 5.37322498552637436041865688150, 6.75574963478171206685547455048, 7.53715153686217732863227816235, 8.446377138974293221095131063340, 8.736785677830695573996941291180

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