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Properties

Label 2-1100-55.49-c1-0-4
Degree $2$
Conductor $1100$
Sign $0.0157 - 0.999i$
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  + (−1.59 + 2.18i)3-s + (−2.21 − 3.04i)7-s + (−1.33 − 4.11i)9-s + (2.98 + 1.43i)11-s + (3.10 − 1.01i)13-s + (−3.71 − 1.20i)17-s + (3.49 + 2.54i)19-s + 10.1·21-s + 6.41i·23-s + (3.40 + 1.10i)27-s + (5.25 − 3.82i)29-s + (1.69 + 5.21i)31-s + (−7.90 + 4.25i)33-s + (5.88 + 8.10i)37-s + (−2.73 + 8.41i)39-s + ⋯
L(s)  = 1  + (−0.918 + 1.26i)3-s + (−0.835 − 1.15i)7-s + (−0.445 − 1.37i)9-s + (0.900 + 0.433i)11-s + (0.862 − 0.280i)13-s + (−0.901 − 0.292i)17-s + (0.802 + 0.583i)19-s + 2.22·21-s + 1.33i·23-s + (0.656 + 0.213i)27-s + (0.976 − 0.709i)29-s + (0.304 + 0.937i)31-s + (−1.37 + 0.740i)33-s + (0.968 + 1.33i)37-s + (−0.437 + 1.34i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0157 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0157 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.0157 - 0.999i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ 0.0157 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.002180004\)
\(L(\frac12)\) \(\approx\) \(1.002180004\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.98 - 1.43i)T \)
good3 \( 1 + (1.59 - 2.18i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (2.21 + 3.04i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-3.10 + 1.01i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.71 + 1.20i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.49 - 2.54i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.41iT - 23T^{2} \)
29 \( 1 + (-5.25 + 3.82i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.69 - 5.21i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.88 - 8.10i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.48 + 4.71i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.906iT - 43T^{2} \)
47 \( 1 + (-3.24 + 4.47i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.03 + 0.337i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.36 - 3.89i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.78 - 8.58i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 13.9iT - 67T^{2} \)
71 \( 1 + (3.20 - 9.86i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.16 + 2.97i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.24 + 6.91i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.81 - 1.88i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-11.7 + 3.81i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.15161805679557355972257750909, −9.562738724505341167544631222756, −8.692690198349982729575970554937, −7.35112815262639038044864217430, −6.55213530940724897784976528229, −5.80946999838953427933211823950, −4.72190414389661955336400259639, −3.97905601790816644707351248991, −3.31531936243612195942981185346, −1.05656681880440226952307751863, 0.63702293679288937654265022796, 1.97924069402771117950763030014, 3.12759573923216710057621965115, 4.58619457190579469614699186030, 5.81944172322229910179413394164, 6.38087523704013316567583324812, 6.70104610608190086508830630930, 7.949862650353221450216839289460, 8.867777949407523692821564903686, 9.409744216588244714297382504644

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