↑

Properties

Label 2-5544-21.20-c1-0-18
Degree $2$
Conductor $5544$
Sign $-0.598 - 0.800i$
Analytic cond. $44.2690$
Root an. cond. $6.65350$
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  + 2.21·5-s + (−0.815 + 2.51i)7-s − i·11-s − 2.68i·13-s − 1.38·17-s + 2.79i·19-s + 7.93i·23-s − 0.0923·25-s + 0.775i·29-s + 0.373i·31-s + (−1.80 + 5.57i)35-s + 2.91·37-s − 10.7·41-s − 9.74·43-s + 8.76·47-s + ⋯
L(s)  = 1  + 0.990·5-s + (−0.308 + 0.951i)7-s − 0.301i·11-s − 0.744i·13-s − 0.336·17-s + 0.640i·19-s + 1.65i·23-s − 0.0184·25-s + 0.144i·29-s + 0.0671i·31-s + (−0.305 + 0.942i)35-s + 0.479·37-s − 1.68·41-s − 1.48·43-s + 1.27·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.598 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5544\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.598 - 0.800i$
Analytic conductor: \(44.2690\)
Root analytic conductor: \(6.65350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5544} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5544,\ (\ :1/2),\ -0.598 - 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.326020345\)
\(L(\frac12)\) \(\approx\) \(1.326020345\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.815 - 2.51i)T \)
11 \( 1 + iT \)
good5 \( 1 - 2.21T + 5T^{2} \)
13 \( 1 + 2.68iT - 13T^{2} \)
17 \( 1 + 1.38T + 17T^{2} \)
19 \( 1 - 2.79iT - 19T^{2} \)
23 \( 1 - 7.93iT - 23T^{2} \)
29 \( 1 - 0.775iT - 29T^{2} \)
31 \( 1 - 0.373iT - 31T^{2} \)
37 \( 1 - 2.91T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 9.74T + 43T^{2} \)
47 \( 1 - 8.76T + 47T^{2} \)
53 \( 1 - 0.694iT - 53T^{2} \)
59 \( 1 + 3.57T + 59T^{2} \)
61 \( 1 - 2.84iT - 61T^{2} \)
67 \( 1 - 0.750T + 67T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 - 3.43iT - 73T^{2} \)
79 \( 1 - 7.75T + 79T^{2} \)
83 \( 1 - 0.471T + 83T^{2} \)
89 \( 1 + 5.92T + 89T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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

−8.472334433460144406329533871662, −7.78295487163447354014747078070, −6.83861539645682322150027762261, −6.11087382899961957062443349462, −5.52892225093221842616962013182, −5.13857587522312449557575996902, −3.79704286841515710155606487458, −3.06202904745991841065722482850, −2.19680568551643105707500329561, −1.37031074684738515286576412683, 0.31843057161067718221061527458, 1.61028755112937076939812069313, 2.36994813624845021837626332335, 3.36731499263237222261253732321, 4.37583005838889846242330334663, 4.83032367406472269669423799181, 5.87144030441238635922419926958, 6.70547204350357886193004195157, 6.83739854501121901187925291706, 7.895623399472509517009058699071

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