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Properties

Label 2-1100-11.5-c1-0-3
Degree $2$
Conductor $1100$
Sign $-0.781 - 0.623i$
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  + (1.64 + 1.19i)3-s + (−3.29 + 2.39i)7-s + (0.343 + 1.05i)9-s + (0.00208 + 3.31i)11-s + (−1.80 − 5.56i)13-s + (−2.13 + 6.56i)17-s + (4.84 + 3.51i)19-s − 8.26·21-s − 1.37·23-s + (1.18 − 3.64i)27-s + (−6.12 + 4.44i)29-s + (1.25 + 3.85i)31-s + (−3.94 + 5.44i)33-s + (−7.98 + 5.80i)37-s + (3.66 − 11.2i)39-s + ⋯
L(s)  = 1  + (0.947 + 0.688i)3-s + (−1.24 + 0.905i)7-s + (0.114 + 0.352i)9-s + (0.000630 + 0.999i)11-s + (−0.501 − 1.54i)13-s + (−0.517 + 1.59i)17-s + (1.11 + 0.806i)19-s − 1.80·21-s − 0.287·23-s + (0.227 − 0.700i)27-s + (−1.13 + 0.825i)29-s + (0.225 + 0.693i)31-s + (−0.687 + 0.947i)33-s + (−1.31 + 0.954i)37-s + (0.587 − 1.80i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.781 - 0.623i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ -0.781 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.338974533\)
\(L(\frac12)\) \(\approx\) \(1.338974533\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.00208 - 3.31i)T \)
good3 \( 1 + (-1.64 - 1.19i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (3.29 - 2.39i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.80 + 5.56i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.13 - 6.56i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.84 - 3.51i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.37T + 23T^{2} \)
29 \( 1 + (6.12 - 4.44i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.25 - 3.85i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.98 - 5.80i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.79 + 1.30i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.49T + 43T^{2} \)
47 \( 1 + (3.50 + 2.54i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.83 - 8.71i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.00 + 5.09i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.37 + 4.24i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.67T + 67T^{2} \)
71 \( 1 + (-1.91 + 5.89i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.80 + 2.04i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.24 - 3.82i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.55 - 7.86i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 + (0.463 + 1.42i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.04657745332939971612554444156, −9.439212401100403283471316407982, −8.635423335338362254126501420583, −7.954932402965466476413861247922, −6.85706946563766720476002201765, −5.86882236207494109491085717529, −5.02901653861688298705586146075, −3.59297774619743882052124664699, −3.23594670269162859809890333702, −2.04441865776561578599086064488, 0.49055501578878481584953446202, 2.17149034660944443814401326299, 3.10096110736859775747037273841, 3.95209372797537468208055550028, 5.21979225152361709584349738477, 6.56530924741580249390146078851, 7.09699473492119564452068954524, 7.66278522706977007284560163238, 8.923191246473913018698864270254, 9.313803066029419150519774517033

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