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Properties

Label 2-1100-11.3-c1-0-16
Degree $2$
Conductor $1100$
Sign $0.204 + 0.978i$
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  + (0.321 + 0.989i)3-s + (0.699 − 2.15i)7-s + (1.55 − 1.12i)9-s + (−2.50 − 2.16i)11-s + (−0.217 + 0.157i)13-s + (−4.79 − 3.48i)17-s + (−1.77 − 5.46i)19-s + 2.35·21-s − 2.92·23-s + (4.13 + 3.00i)27-s + (2.00 − 6.16i)29-s + (−0.202 + 0.146i)31-s + (1.34 − 3.17i)33-s + (−3.18 + 9.79i)37-s + (−0.225 − 0.164i)39-s + ⋯
L(s)  = 1  + (0.185 + 0.571i)3-s + (0.264 − 0.813i)7-s + (0.517 − 0.375i)9-s + (−0.756 − 0.654i)11-s + (−0.0601 + 0.0437i)13-s + (−1.16 − 0.844i)17-s + (−0.407 − 1.25i)19-s + 0.513·21-s − 0.610·23-s + (0.796 + 0.578i)27-s + (0.371 − 1.14i)29-s + (−0.0363 + 0.0263i)31-s + (0.233 − 0.553i)33-s + (−0.523 + 1.61i)37-s + (−0.0361 − 0.0262i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.204 + 0.978i$
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: \((2,\ 1100,\ (\ :1/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.349020303\)
\(L(\frac12)\) \(\approx\) \(1.349020303\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.50 + 2.16i)T \)
good3 \( 1 + (-0.321 - 0.989i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.699 + 2.15i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.217 - 0.157i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.79 + 3.48i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.77 + 5.46i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.92T + 23T^{2} \)
29 \( 1 + (-2.00 + 6.16i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.202 - 0.146i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.18 - 9.79i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.730 + 2.24i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.56T + 43T^{2} \)
47 \( 1 + (0.349 + 1.07i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.25 + 5.99i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.52 + 10.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.73 - 2.71i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.11T + 67T^{2} \)
71 \( 1 + (4.11 + 2.99i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.52 - 7.75i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.71 + 4.88i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-13.2 - 9.65i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.73T + 89T^{2} \)
97 \( 1 + (-3.65 + 2.65i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.828381758482993571199217041343, −8.850116477921805230217408626441, −8.166978346734256910933914153489, −7.07609766006653730819255393288, −6.50640285503763083519348132425, −5.07337401372084203913481600088, −4.47223481188656011912914839372, −3.51201877451047481420628366329, −2.34854775999596368887724224267, −0.56235989430303020513116724707, 1.77722994618713293408940274423, 2.35614257417165989008524587201, 3.89943946449292961670933999378, 4.90566058467562133030035121648, 5.82110556296640175344503208846, 6.77679008099699964515567885934, 7.61576379018627170302788119060, 8.342342903252350967602849441752, 9.001831118150046083668826570749, 10.28451455486558671019876222447

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