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Properties

Label 2-1100-55.32-c1-0-10
Degree $2$
Conductor $1100$
Sign $0.853 + 0.520i$
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  + (0.323 − 0.323i)3-s + (−0.901 + 0.901i)7-s + 2.79i·9-s + (−1.79 − 2.79i)11-s + (−2.51 − 2.51i)13-s + (5.22 − 5.22i)17-s + 5·19-s + 0.582i·21-s + (2.83 − 2.83i)23-s + (1.87 + 1.87i)27-s + 8.37·29-s + 4.58·31-s + (−1.48 − 0.323i)33-s + (4.96 + 4.96i)37-s − 1.62·39-s + ⋯
L(s)  = 1  + (0.186 − 0.186i)3-s + (−0.340 + 0.340i)7-s + 0.930i·9-s + (−0.540 − 0.841i)11-s + (−0.698 − 0.698i)13-s + (1.26 − 1.26i)17-s + 1.14·19-s + 0.127i·21-s + (0.592 − 0.592i)23-s + (0.360 + 0.360i)27-s + 1.55·29-s + 0.823·31-s + (−0.257 − 0.0562i)33-s + (0.816 + 0.816i)37-s − 0.260·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.853 + 0.520i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ 0.853 + 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.635328935\)
\(L(\frac12)\) \(\approx\) \(1.635328935\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 + 2.79i)T \)
good3 \( 1 + (-0.323 + 0.323i)T - 3iT^{2} \)
7 \( 1 + (0.901 - 0.901i)T - 7iT^{2} \)
13 \( 1 + (2.51 + 2.51i)T + 13iT^{2} \)
17 \( 1 + (-5.22 + 5.22i)T - 17iT^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (-2.83 + 2.83i)T - 23iT^{2} \)
29 \( 1 - 8.37T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 + (-4.96 - 4.96i)T + 37iT^{2} \)
41 \( 1 - 5iT - 41T^{2} \)
43 \( 1 + (6.83 + 6.83i)T + 43iT^{2} \)
47 \( 1 + (0.578 + 0.578i)T + 47iT^{2} \)
53 \( 1 + (-7.09 + 7.09i)T - 53iT^{2} \)
59 \( 1 - 8.58iT - 59T^{2} \)
61 \( 1 + 7.79iT - 61T^{2} \)
67 \( 1 + (6.19 + 6.19i)T + 67iT^{2} \)
71 \( 1 + 0.417T + 71T^{2} \)
73 \( 1 + (0.901 + 0.901i)T + 73iT^{2} \)
79 \( 1 - 7.20T + 79T^{2} \)
83 \( 1 + (5.22 + 5.22i)T + 83iT^{2} \)
89 \( 1 + 4.79iT - 89T^{2} \)
97 \( 1 + (-3.35 - 3.35i)T + 97iT^{2} \)
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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

−10.00058461472580234417997629462, −8.880177568251259320951975009109, −7.996779683301576404150993542776, −7.52979600611865278945682643290, −6.43080479709935565452880169689, −5.28850876691964703120027854930, −4.91781490046104132841559445726, −3.02763377765856675825723540275, −2.77599928629211002028491996623, −0.867601483230896735814702437595, 1.18110071039350545922711705183, 2.74849544593921317877954758797, 3.69230020363874268990680757965, 4.63690977939394535058905314651, 5.66920303469979311569479894703, 6.68286951291388117178968210085, 7.39810591241107100171663770751, 8.254382284867585044008875215352, 9.323119989708159339906337648490, 9.915617985417953743193785087796

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