Properties

Label 2-1100-5.4-c1-0-3
Degree $2$
Conductor $1100$
Sign $-0.894 - 0.447i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.79i·3-s + 4.79i·7-s − 0.208·9-s − 11-s i·13-s + 3.79i·17-s + 2.58·19-s − 8.58·21-s + 0.791i·23-s + 5.00i·27-s − 2.20·29-s + 0.582·31-s − 1.79i·33-s − 6.58i·37-s + 1.79·39-s + ⋯
L(s)  = 1  + 1.03i·3-s + 1.81i·7-s − 0.0695·9-s − 0.301·11-s − 0.277i·13-s + 0.919i·17-s + 0.592·19-s − 1.87·21-s + 0.164i·23-s + 0.962i·27-s − 0.410·29-s + 0.104·31-s − 0.311i·33-s − 1.08i·37-s + 0.286·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.390213349\)
\(L(\frac12)\) \(\approx\) \(1.390213349\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 - 1.79iT - 3T^{2} \)
7 \( 1 - 4.79iT - 7T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 3.79iT - 17T^{2} \)
19 \( 1 - 2.58T + 19T^{2} \)
23 \( 1 - 0.791iT - 23T^{2} \)
29 \( 1 + 2.20T + 29T^{2} \)
31 \( 1 - 0.582T + 31T^{2} \)
37 \( 1 + 6.58iT - 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 - 10.5iT - 47T^{2} \)
53 \( 1 + 2.37iT - 53T^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 - 8.79T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 16.7T + 71T^{2} \)
73 \( 1 + 3.20iT - 73T^{2} \)
79 \( 1 + 16.5T + 79T^{2} \)
83 \( 1 - 12.9iT - 83T^{2} \)
89 \( 1 + 3.79T + 89T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.08141520383487296141256445795, −9.402893901059134096184535663764, −8.740041136030264167055978697142, −7.998960736920526174337529527975, −6.75360959831682662220650797476, −5.53162913909998488141051766085, −5.32207774160311750296445549845, −4.04929585101012358080694340617, −3.09275382058044285948534933225, −1.95458200588052628896219001414, 0.62501305905652533546536089243, 1.66072238751476767802702823504, 3.12619379043241550266004382526, 4.21864141573697743138806802541, 5.13668151145891497198652472129, 6.55553196354825656720438605706, 7.02510777925570319407286241155, 7.64300291594859008223193863154, 8.390910479282867117532091184312, 9.770589560513343747675033790546

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