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Properties

Label 2-1100-11.10-c2-0-29
Degree $2$
Conductor $1100$
Sign $-0.208 + 0.977i$
Analytic cond. $29.9728$
Root an. cond. $5.47474$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.81·3-s + 4.15i·7-s − 5.70·9-s + (−2.29 + 10.7i)11-s − 23.6i·13-s + 4.15i·17-s − 29.0i·19-s + 7.54i·21-s − 24.3·23-s − 26.7·27-s − 13.9i·29-s + 9.50·31-s + (−4.17 + 19.5i)33-s + 46.8·37-s − 43.0i·39-s + ⋯
L(s)  = 1  + 0.605·3-s + 0.593i·7-s − 0.633·9-s + (−0.208 + 0.977i)11-s − 1.82i·13-s + 0.244i·17-s − 1.52i·19-s + 0.359i·21-s − 1.05·23-s − 0.988·27-s − 0.481i·29-s + 0.306·31-s + (−0.126 + 0.592i)33-s + 1.26·37-s − 1.10i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.208 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.208 + 0.977i$
Analytic conductor: \(29.9728\)
Root analytic conductor: \(5.47474\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1),\ -0.208 + 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.301017888\)
\(L(\frac12)\) \(\approx\) \(1.301017888\)
\(L(2)\) 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 \( 1 + (2.29 - 10.7i)T \)
good3 \( 1 - 1.81T + 9T^{2} \)
7 \( 1 - 4.15iT - 49T^{2} \)
13 \( 1 + 23.6iT - 169T^{2} \)
17 \( 1 - 4.15iT - 289T^{2} \)
19 \( 1 + 29.0iT - 361T^{2} \)
23 \( 1 + 24.3T + 529T^{2} \)
29 \( 1 + 13.9iT - 841T^{2} \)
31 \( 1 - 9.50T + 961T^{2} \)
37 \( 1 - 46.8T + 1.36e3T^{2} \)
41 \( 1 + 58.1iT - 1.68e3T^{2} \)
43 \( 1 + 62.7iT - 1.84e3T^{2} \)
47 \( 1 + 18.1T + 2.20e3T^{2} \)
53 \( 1 - 62.8T + 2.80e3T^{2} \)
59 \( 1 + 60.8T + 3.48e3T^{2} \)
61 \( 1 + 72.0iT - 3.72e3T^{2} \)
67 \( 1 - 62.6T + 4.48e3T^{2} \)
71 \( 1 + 113.T + 5.04e3T^{2} \)
73 \( 1 - 126. iT - 5.32e3T^{2} \)
79 \( 1 + 70.9iT - 6.24e3T^{2} \)
83 \( 1 - 87.6iT - 6.88e3T^{2} \)
89 \( 1 - 26.1T + 7.92e3T^{2} \)
97 \( 1 + 105.T + 9.40e3T^{2} \)
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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

−9.370303526597621487409224286204, −8.533737958713567690080941671485, −7.940500730397244418094071635664, −7.09373154651769823329405939861, −5.85107090907384180932142325850, −5.26049082996420837378816796099, −4.04129014539361302528828731510, −2.83732310258915402802102824996, −2.26270514740368013242010839296, −0.35363581643071997888794428015, 1.40773854337369503923377591105, 2.65049085898688059592025744018, 3.69447704208788912677564331595, 4.45316388596115033607149345838, 5.83898976023079382971943564472, 6.44337601379305920834893666758, 7.65564622534611888872377444060, 8.208036214685599241826782687140, 9.060494053881469750688228795743, 9.749195090413796082217788548389

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