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Properties

Label 2-1100-55.32-c1-0-9
Degree $2$
Conductor $1100$
Sign $0.940 + 0.339i$
Analytic cond. $8.78354$
Root an. cond. $2.96370$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.54 − 1.54i)3-s + (−2.77 + 2.77i)7-s − 1.79i·9-s + (2.79 − 1.79i)11-s + (4.96 + 4.96i)13-s + (3.35 − 3.35i)17-s − 5·19-s + 8.58i·21-s + (6.51 − 6.51i)23-s + (1.87 + 1.87i)27-s + 5.37·29-s − 4.58·31-s + (1.54 − 7.09i)33-s + (2.51 + 2.51i)37-s + 15.3·39-s + ⋯
L(s)  = 1  + (0.893 − 0.893i)3-s + (−1.04 + 1.04i)7-s − 0.597i·9-s + (0.841 − 0.540i)11-s + (1.37 + 1.37i)13-s + (0.812 − 0.812i)17-s − 1.14·19-s + 1.87i·21-s + (1.35 − 1.35i)23-s + (0.360 + 0.360i)27-s + 0.997·29-s − 0.823·31-s + (0.269 − 1.23i)33-s + (0.413 + 0.413i)37-s + 2.46·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.940 + 0.339i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ 0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.209874813\)
\(L(\frac12)\) \(\approx\) \(2.209874813\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.79 + 1.79i)T \)
good3 \( 1 + (-1.54 + 1.54i)T - 3iT^{2} \)
7 \( 1 + (2.77 - 2.77i)T - 7iT^{2} \)
13 \( 1 + (-4.96 - 4.96i)T + 13iT^{2} \)
17 \( 1 + (-3.35 + 3.35i)T - 17iT^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-6.51 + 6.51i)T - 23iT^{2} \)
29 \( 1 - 5.37T + 29T^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 + (-2.51 - 2.51i)T + 37iT^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 + (-4.38 - 4.38i)T + 43iT^{2} \)
47 \( 1 + (-4.32 - 4.32i)T + 47iT^{2} \)
53 \( 1 + (1.48 - 1.48i)T - 53iT^{2} \)
59 \( 1 + 0.582iT - 59T^{2} \)
61 \( 1 - 3.20iT - 61T^{2} \)
67 \( 1 + (1.29 + 1.29i)T + 67iT^{2} \)
71 \( 1 + 9.58T + 71T^{2} \)
73 \( 1 + (2.77 + 2.77i)T + 73iT^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + (3.35 + 3.35i)T + 83iT^{2} \)
89 \( 1 + 0.208iT - 89T^{2} \)
97 \( 1 + (5.22 + 5.22i)T + 97iT^{2} \)
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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

−9.342007297010278284972666291105, −8.840239435955138616235076281444, −8.520954343612087015443574251708, −7.18349424483879009776613692516, −6.49830035342139951921729290406, −5.96223838774482253731542977515, −4.40135402405003763641310088845, −3.22739168328416679066793297251, −2.51425490901890557216284398334, −1.24493714279411688855174624387, 1.14637937678421264801937488976, 3.06114961288241012201038029965, 3.65781298422486512715748108483, 4.22451377070499709436004616697, 5.67983143354281670014791749600, 6.57106627077436690094174593576, 7.49987119066762358263688505240, 8.476866009062108500908848400183, 9.109519050076761732889807460372, 9.963933131353683367810348222943

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