Properties

Label 2-1100-5.4-c3-0-22
Degree $2$
Conductor $1100$
Sign $0.447 - 0.894i$
Analytic cond. $64.9021$
Root an. cond. $8.05618$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.424i·3-s + 34.5i·7-s + 26.8·9-s + 11·11-s + 1.15i·13-s − 60.3i·17-s + 92.4·19-s − 14.6·21-s − 53.7i·23-s + 22.8i·27-s + 275.·29-s + 195.·31-s + 4.66i·33-s + 283. i·37-s − 0.488·39-s + ⋯
L(s)  = 1  + 0.0816i·3-s + 1.86i·7-s + 0.993·9-s + 0.301·11-s + 0.0245i·13-s − 0.860i·17-s + 1.11·19-s − 0.152·21-s − 0.487i·23-s + 0.162i·27-s + 1.76·29-s + 1.13·31-s + 0.0246i·33-s + 1.26i·37-s − 0.00200·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(64.9021\)
Root analytic conductor: \(8.05618\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.543148517\)
\(L(\frac12)\) \(\approx\) \(2.543148517\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 - 11T \)
good3 \( 1 - 0.424iT - 27T^{2} \)
7 \( 1 - 34.5iT - 343T^{2} \)
13 \( 1 - 1.15iT - 2.19e3T^{2} \)
17 \( 1 + 60.3iT - 4.91e3T^{2} \)
19 \( 1 - 92.4T + 6.85e3T^{2} \)
23 \( 1 + 53.7iT - 1.21e4T^{2} \)
29 \( 1 - 275.T + 2.43e4T^{2} \)
31 \( 1 - 195.T + 2.97e4T^{2} \)
37 \( 1 - 283. iT - 5.06e4T^{2} \)
41 \( 1 + 70.3T + 6.89e4T^{2} \)
43 \( 1 + 96.9iT - 7.95e4T^{2} \)
47 \( 1 + 510. iT - 1.03e5T^{2} \)
53 \( 1 - 298. iT - 1.48e5T^{2} \)
59 \( 1 - 261.T + 2.05e5T^{2} \)
61 \( 1 + 418.T + 2.26e5T^{2} \)
67 \( 1 + 304. iT - 3.00e5T^{2} \)
71 \( 1 - 641.T + 3.57e5T^{2} \)
73 \( 1 - 182. iT - 3.89e5T^{2} \)
79 \( 1 + 806.T + 4.93e5T^{2} \)
83 \( 1 - 926. iT - 5.71e5T^{2} \)
89 \( 1 + 1.24e3T + 7.04e5T^{2} \)
97 \( 1 - 1.80e3iT - 9.12e5T^{2} \)
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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

−9.663867641574440113671232755867, −8.796264714929233851582428169493, −8.169766671375708812355843136738, −6.99772414881926537455535161769, −6.28780195314576827776076684284, −5.23689142229839382364136220438, −4.62443052627659266776148349062, −3.17945779584664780713002033681, −2.35766918284610418828707694241, −1.07445319328066040781097276063, 0.78211371930343624794275052940, 1.48349086701294794566376495018, 3.20856304711560725660707118327, 4.13386221253302776357025954390, 4.70893609426036236708379272613, 6.13932812874715321688748180278, 6.98443054675023967998840586717, 7.52852995524753730211870474488, 8.329867152945441395726361581451, 9.634454566894909661208757678873

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