↑

Properties

Label 2-2200-440.379-c0-0-4
Degree $2$
Conductor $2200$
Sign $0.405 + 0.913i$
Analytic cond. $1.09794$
Root an. cond. $1.04782$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 0.309i)2-s + (0.363 − 0.5i)3-s + (0.809 − 0.587i)4-s + (0.190 − 0.587i)6-s + (0.587 − 0.809i)8-s + (0.190 + 0.587i)9-s + (−0.809 − 0.587i)11-s − 0.618i·12-s + (0.309 − 0.951i)16-s + (1.53 + 0.5i)17-s + (0.363 + 0.5i)18-s + (−1.30 − 0.951i)19-s + (−0.951 − 0.309i)22-s + (−0.190 − 0.587i)24-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.363 − 0.5i)3-s + (0.809 − 0.587i)4-s + (0.190 − 0.587i)6-s + (0.587 − 0.809i)8-s + (0.190 + 0.587i)9-s + (−0.809 − 0.587i)11-s − 0.618i·12-s + (0.309 − 0.951i)16-s + (1.53 + 0.5i)17-s + (0.363 + 0.5i)18-s + (−1.30 − 0.951i)19-s + (−0.951 − 0.309i)22-s + (−0.190 − 0.587i)24-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.405 + 0.913i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.405 + 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.376512776\)
\(L(\frac12)\) \(\approx\) \(2.376512776\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
11 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - 1.61iT - T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + 0.618T + T^{2} \)
97 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
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

−9.041470433010621992374004851986, −8.072478205808267887392512659716, −7.57578822517362609842495788375, −6.66193645074692795891006629917, −5.84434458088726782632134745129, −5.09114328119638413328249701583, −4.24539648042962758229736058678, −3.12926693082698504100312506264, −2.46438906711067287868590713922, −1.36313367185505042808855586698, 1.85137598401519212333277465356, 3.01654145332740037315510272537, 3.68539162722712565244847232563, 4.55271517083017078413623007110, 5.30443269136959957195074968011, 6.16132981208331888936752207080, 6.95516800934519639484972906000, 7.83545371738352375482629250406, 8.359141095291208795519539612692, 9.471636376126101767340265540000

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