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Properties

Label 2-1100-11.3-c1-0-10
Degree $2$
Conductor $1100$
Sign $0.305 + 0.952i$
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  + (−0.655 − 2.01i)3-s + (−0.0946 + 0.291i)7-s + (−1.21 + 0.884i)9-s + (2.72 + 1.89i)11-s + (2.68 − 1.95i)13-s + (4.58 + 3.33i)17-s + (0.464 + 1.43i)19-s + 0.650·21-s + 0.343·23-s + (−2.56 − 1.86i)27-s + (2.15 − 6.64i)29-s + (4.80 − 3.49i)31-s + (2.04 − 6.73i)33-s + (−1.63 + 5.04i)37-s + (−5.70 − 4.14i)39-s + ⋯
L(s)  = 1  + (−0.378 − 1.16i)3-s + (−0.0357 + 0.110i)7-s + (−0.405 + 0.294i)9-s + (0.820 + 0.571i)11-s + (0.745 − 0.541i)13-s + (1.11 + 0.808i)17-s + (0.106 + 0.328i)19-s + 0.141·21-s + 0.0716·23-s + (−0.494 − 0.359i)27-s + (0.400 − 1.23i)29-s + (0.863 − 0.627i)31-s + (0.355 − 1.17i)33-s + (−0.269 + 0.828i)37-s + (−0.912 − 0.663i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.305 + 0.952i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ 0.305 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.585179560\)
\(L(\frac12)\) \(\approx\) \(1.585179560\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.72 - 1.89i)T \)
good3 \( 1 + (0.655 + 2.01i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.0946 - 0.291i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.68 + 1.95i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.58 - 3.33i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.464 - 1.43i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.343T + 23T^{2} \)
29 \( 1 + (-2.15 + 6.64i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.80 + 3.49i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.63 - 5.04i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.25 + 6.94i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.16T + 43T^{2} \)
47 \( 1 + (-1.94 - 5.98i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.63 + 6.27i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.590 - 1.81i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.27 + 6.01i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + (9.03 + 6.56i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.792 + 2.43i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.95 - 1.42i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.66 - 2.66i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.46T + 89T^{2} \)
97 \( 1 + (-11.1 + 8.06i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.808263836758977390091610651068, −8.670093597071071618875204834682, −7.895417638881243116372493086438, −7.21450198508615750727351336361, −6.17788058082874331771622464128, −5.88458140622294164381659235024, −4.43137833792860954274655016746, −3.35901580434512293620894870462, −1.90091953742804811846662271762, −0.945643676080371526628535153423, 1.20661854758882257988801991683, 3.10949104486261802436191706645, 3.88843613439065819485386687645, 4.80897603297814658046333001544, 5.60149349042768055252955962635, 6.56536836300389476244519855497, 7.47350595447488514470596502388, 8.799094215463522380146492949877, 9.116231773349160808162984714700, 10.22496329937138313682975009989

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