Properties

Label 2-11e2-1.1-c3-0-9
Degree $2$
Conductor $121$
Sign $1$
Analytic cond. $7.13923$
Root an. cond. $2.67193$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s − 8·4-s + 18·5-s + 37·9-s − 64·12-s + 144·15-s + 64·16-s − 144·20-s − 108·23-s + 199·25-s + 80·27-s + 340·31-s − 296·36-s − 434·37-s + 666·45-s − 36·47-s + 512·48-s − 343·49-s − 738·53-s − 720·59-s − 1.15e3·60-s − 512·64-s − 416·67-s − 864·69-s + 612·71-s + 1.59e3·75-s + 1.15e3·80-s + ⋯
L(s)  = 1  + 1.53·3-s − 4-s + 1.60·5-s + 1.37·9-s − 1.53·12-s + 2.47·15-s + 16-s − 1.60·20-s − 0.979·23-s + 1.59·25-s + 0.570·27-s + 1.96·31-s − 1.37·36-s − 1.92·37-s + 2.20·45-s − 0.111·47-s + 1.53·48-s − 49-s − 1.91·53-s − 1.58·59-s − 2.47·60-s − 64-s − 0.758·67-s − 1.50·69-s + 1.02·71-s + 2.45·75-s + 1.60·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $1$
Analytic conductor: \(7.13923\)
Root analytic conductor: \(2.67193\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.657507929\)
\(L(\frac12)\) \(\approx\) \(2.657507929\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + p^{3} T^{2} \)
3 \( 1 - 8 T + p^{3} T^{2} \)
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

−13.48719827604743524387288521679, −12.45865932695564573767939355280, −10.26170987457387885185440199249, −9.639403516109659497303370707326, −8.860604594052153684972063398629, −7.967102512099203374505555448222, −6.22995073947286370673113788833, −4.78439786303315172184530317259, −3.20836200081001826225041312589, −1.78483866409193609840635518996, 1.78483866409193609840635518996, 3.20836200081001826225041312589, 4.78439786303315172184530317259, 6.22995073947286370673113788833, 7.967102512099203374505555448222, 8.860604594052153684972063398629, 9.639403516109659497303370707326, 10.26170987457387885185440199249, 12.45865932695564573767939355280, 13.48719827604743524387288521679

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