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Properties

Label 2-1100-11.3-c1-0-7
Degree $2$
Conductor $1100$
Sign $0.333 - 0.942i$
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  + (0.616 + 1.89i)3-s + (0.139 − 0.430i)7-s + (−0.797 + 0.579i)9-s + (1.01 + 3.15i)11-s + (5.32 − 3.87i)13-s + (−0.959 − 0.696i)17-s + (0.575 + 1.77i)19-s + 0.903·21-s + 5.37·23-s + (3.25 + 2.36i)27-s + (−2.35 + 7.25i)29-s + (−2.83 + 2.05i)31-s + (−5.36 + 3.87i)33-s + (1.78 − 5.50i)37-s + (10.6 + 7.72i)39-s + ⋯
L(s)  = 1  + (0.356 + 1.09i)3-s + (0.0528 − 0.162i)7-s + (−0.265 + 0.193i)9-s + (0.306 + 0.951i)11-s + (1.47 − 1.07i)13-s + (−0.232 − 0.168i)17-s + (0.132 + 0.406i)19-s + 0.197·21-s + 1.11·23-s + (0.626 + 0.454i)27-s + (−0.437 + 1.34i)29-s + (−0.508 + 0.369i)31-s + (−0.934 + 0.675i)33-s + (0.294 − 0.904i)37-s + (1.70 + 1.23i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.333 - 0.942i$
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: \((2,\ 1100,\ (\ :1/2),\ 0.333 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.027484937\)
\(L(\frac12)\) \(\approx\) \(2.027484937\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.01 - 3.15i)T \)
good3 \( 1 + (-0.616 - 1.89i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.139 + 0.430i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-5.32 + 3.87i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.959 + 0.696i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.575 - 1.77i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.37T + 23T^{2} \)
29 \( 1 + (2.35 - 7.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.83 - 2.05i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.78 + 5.50i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.103 - 0.317i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + (-1.98 - 6.09i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.85 + 1.34i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.449 + 1.38i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.22 + 5.25i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 4.37T + 67T^{2} \)
71 \( 1 + (-8.91 - 6.47i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.09 + 3.36i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.46 + 1.79i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.95 + 5.78i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + (-11.2 + 8.14i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.03618256127712093917466993124, −9.160259433824650837182874010031, −8.678504594239662058413322518028, −7.58505125601291003457702591486, −6.71176678924710145473234029711, −5.55590767046173357915791620526, −4.72292847622825239641666668641, −3.77094475484248375579283793374, −3.10733758403803069709267989494, −1.39739620367811548993509045386, 1.03548185214962837627981561849, 2.05948770301400677078185571941, 3.28937156521283294639204650881, 4.32136561355290715626797695441, 5.68673039256893310767262011345, 6.51070959804564568603951208413, 7.06812684004826972520958144585, 8.198100081661643437888171141825, 8.651394545916483888951272032762, 9.434270840305318780711532454578

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