↑

Properties

Label 2-1100-55.49-c1-0-12
Degree $2$
Conductor $1100$
Sign $0.362 + 0.932i$
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  + (−1.76 + 2.42i)3-s + (0.992 + 1.36i)7-s + (−1.85 − 5.70i)9-s + (2.77 − 1.81i)11-s + (−3.52 + 1.14i)13-s + (−3.68 − 1.19i)17-s + (−5.21 − 3.78i)19-s − 5.06·21-s − 7.43i·23-s + (8.56 + 2.78i)27-s + (−2.55 + 1.85i)29-s + (0.787 + 2.42i)31-s + (−0.472 + 9.93i)33-s + (3.84 + 5.29i)37-s + (3.44 − 10.5i)39-s + ⋯
L(s)  = 1  + (−1.01 + 1.40i)3-s + (0.375 + 0.516i)7-s + (−0.618 − 1.90i)9-s + (0.835 − 0.548i)11-s + (−0.978 + 0.318i)13-s + (−0.892 − 0.290i)17-s + (−1.19 − 0.868i)19-s − 1.10·21-s − 1.55i·23-s + (1.64 + 0.535i)27-s + (−0.475 + 0.345i)29-s + (0.141 + 0.435i)31-s + (−0.0821 + 1.73i)33-s + (0.632 + 0.871i)37-s + (0.550 − 1.69i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.362 + 0.932i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ 0.362 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3796325214\)
\(L(\frac12)\) \(\approx\) \(0.3796325214\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.77 + 1.81i)T \)
good3 \( 1 + (1.76 - 2.42i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-0.992 - 1.36i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (3.52 - 1.14i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.68 + 1.19i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.21 + 3.78i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 7.43iT - 23T^{2} \)
29 \( 1 + (2.55 - 1.85i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.787 - 2.42i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.84 - 5.29i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.93 + 5.04i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.49iT - 43T^{2} \)
47 \( 1 + (-2.49 + 3.44i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (9.24 - 3.00i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.33 - 1.69i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.93 + 5.94i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 10.5iT - 67T^{2} \)
71 \( 1 + (-2.85 + 8.78i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.15 - 2.96i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-5.23 - 16.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-11.4 - 3.71i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + (6.38 - 2.07i)T + (78.4 - 57.0i)T^{2} \)
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

−9.693330719022740796626306193173, −9.047040871555489546197138232231, −8.432792786923607508819249373100, −6.75175849328860016174862561954, −6.32976349976851045634766597160, −5.06668553948921904542283763830, −4.70643209726189094825942322291, −3.76150894140560720982742529039, −2.35962627241924083585769884260, −0.19423209079384926328274643690, 1.36932997748062101868180957170, 2.20225015923127444032732688956, 4.02935235992499123047967980847, 4.98663243974612628044581341132, 6.04113295837947684577842621125, 6.60770076492131700771318481765, 7.54530759203751606814890064442, 7.88098590361100418890911405079, 9.194275484857993738181428624151, 10.17655075175626212018078706475

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