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Properties

Label 2-1100-5.2-c2-0-15
Degree $2$
Conductor $1100$
Sign $0.899 - 0.437i$
Analytic cond. $29.9728$
Root an. cond. $5.47474$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.56 − 2.56i)3-s + (9.58 + 9.58i)7-s − 4.16i·9-s + 3.31·11-s + (−7.01 + 7.01i)13-s + (3.73 + 3.73i)17-s − 6.94i·19-s + 49.1·21-s + (−10.2 + 10.2i)23-s + (12.4 + 12.4i)27-s + 49.3i·29-s + 0.909·31-s + (8.50 − 8.50i)33-s + (−1.17 − 1.17i)37-s + 35.9i·39-s + ⋯
L(s)  = 1  + (0.855 − 0.855i)3-s + (1.36 + 1.36i)7-s − 0.462i·9-s + 0.301·11-s + (−0.539 + 0.539i)13-s + (0.219 + 0.219i)17-s − 0.365i·19-s + 2.34·21-s + (−0.444 + 0.444i)23-s + (0.459 + 0.459i)27-s + 1.70i·29-s + 0.0293·31-s + (0.257 − 0.257i)33-s + (−0.0317 − 0.0317i)37-s + 0.922i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(29.9728\)
Root analytic conductor: \(5.47474\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1),\ 0.899 - 0.437i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.004079353\)
\(L(\frac12)\) \(\approx\) \(3.004079353\)
\(L(2)\) 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 \( 1 - 3.31T \)
good3 \( 1 + (-2.56 + 2.56i)T - 9iT^{2} \)
7 \( 1 + (-9.58 - 9.58i)T + 49iT^{2} \)
13 \( 1 + (7.01 - 7.01i)T - 169iT^{2} \)
17 \( 1 + (-3.73 - 3.73i)T + 289iT^{2} \)
19 \( 1 + 6.94iT - 361T^{2} \)
23 \( 1 + (10.2 - 10.2i)T - 529iT^{2} \)
29 \( 1 - 49.3iT - 841T^{2} \)
31 \( 1 - 0.909T + 961T^{2} \)
37 \( 1 + (1.17 + 1.17i)T + 1.36e3iT^{2} \)
41 \( 1 - 11.7T + 1.68e3T^{2} \)
43 \( 1 + (-30.2 + 30.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (3.10 + 3.10i)T + 2.20e3iT^{2} \)
53 \( 1 + (54.9 - 54.9i)T - 2.80e3iT^{2} \)
59 \( 1 + 51.2iT - 3.48e3T^{2} \)
61 \( 1 + 71.9T + 3.72e3T^{2} \)
67 \( 1 + (22.3 + 22.3i)T + 4.48e3iT^{2} \)
71 \( 1 - 10.6T + 5.04e3T^{2} \)
73 \( 1 + (-85.4 + 85.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 0.878iT - 6.24e3T^{2} \)
83 \( 1 + (-82.9 + 82.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 114. iT - 7.92e3T^{2} \)
97 \( 1 + (-42.4 - 42.4i)T + 9.40e3iT^{2} \)
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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

−9.165515664234045180367821468323, −8.971452226317943409997418050122, −7.987549980131317485095624985959, −7.53662728976839854354247869897, −6.47429346865606016639065553447, −5.39243168717465773965055270265, −4.63217850778606076268428457160, −3.13540433787341683571570997348, −2.11870740315482424087786518653, −1.53546947209991434324897590980, 0.853230803686723843254713132600, 2.27168527978522309939094512638, 3.54403981222881886004498560293, 4.30335588402555678494842931412, 4.88725850777894656147273636554, 6.22271679781711021216407541677, 7.52225889921277490046462355799, 7.920485372515652237071651513918, 8.727968134839603199344413295480, 9.821725764033413262293630274675

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