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Properties

Label 2-1100-11.10-c0-0-2
Degree $2$
Conductor $1100$
Sign $1$
Analytic cond. $0.548971$
Root an. cond. $0.740926$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·3-s + 1.99·9-s − 11-s − 1.73·23-s + 1.73·27-s + 31-s − 1.73·33-s − 1.73·37-s + 49-s + 59-s − 1.73·67-s − 2.99·69-s − 71-s + 0.999·81-s − 89-s + 1.73·93-s + 1.73·97-s − 1.99·99-s − 2.99·111-s + 1.73·113-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.99·9-s − 11-s − 1.73·23-s + 1.73·27-s + 31-s − 1.73·33-s − 1.73·37-s + 49-s + 59-s − 1.73·67-s − 2.99·69-s − 71-s + 0.999·81-s − 89-s + 1.73·93-s + 1.73·97-s − 1.99·99-s − 2.99·111-s + 1.73·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.548971\)
Root analytic conductor: \(0.740926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (901, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.709184257\)
\(L(\frac12)\) \(\approx\) \(1.709184257\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 1.73T + T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.73T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + 1.73T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.73T + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 - 1.73T + T^{2} \)
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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

−10.12459472806766885954537818362, −9.042278849217249524231650432152, −8.407203293285622527421481733753, −7.79343182612736339393143561646, −7.06344963719265148844445609685, −5.82434080685729024192167866888, −4.59805343546361649538028565164, −3.67008357008188912795257413478, −2.74958642943566616652444799837, −1.88673636656668013897122898323, 1.88673636656668013897122898323, 2.74958642943566616652444799837, 3.67008357008188912795257413478, 4.59805343546361649538028565164, 5.82434080685729024192167866888, 7.06344963719265148844445609685, 7.79343182612736339393143561646, 8.407203293285622527421481733753, 9.042278849217249524231650432152, 10.12459472806766885954537818362

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