↑

Properties

Label 2-2200-440.107-c0-0-0
Degree $2$
Conductor $2200$
Sign $-0.999 - 0.0128i$
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  + (−0.987 − 0.156i)2-s + (0.951 + 0.309i)4-s + (−0.550 + 0.280i)7-s + (−0.891 − 0.453i)8-s + (−0.587 − 0.809i)9-s + (−0.309 + 0.951i)11-s + (−0.253 + 1.59i)13-s + (0.587 − 0.190i)14-s + (0.809 + 0.587i)16-s + (0.453 + 0.891i)18-s + (−0.363 − 1.11i)19-s + (0.453 − 0.891i)22-s + (−1.34 − 1.34i)23-s + (0.5 − 1.53i)26-s + (−0.610 + 0.0966i)28-s + ⋯
L(s)  = 1  + (−0.987 − 0.156i)2-s + (0.951 + 0.309i)4-s + (−0.550 + 0.280i)7-s + (−0.891 − 0.453i)8-s + (−0.587 − 0.809i)9-s + (−0.309 + 0.951i)11-s + (−0.253 + 1.59i)13-s + (0.587 − 0.190i)14-s + (0.809 + 0.587i)16-s + (0.453 + 0.891i)18-s + (−0.363 − 1.11i)19-s + (0.453 − 0.891i)22-s + (−1.34 − 1.34i)23-s + (0.5 − 1.53i)26-s + (−0.610 + 0.0966i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-0.999 - 0.0128i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ -0.999 - 0.0128i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01048792989\)
\(L(\frac12)\) \(\approx\) \(0.01048792989\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.987 + 0.156i)T \)
5 \( 1 \)
11 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.587 + 0.809i)T^{2} \)
7 \( 1 + (0.550 - 0.280i)T + (0.587 - 0.809i)T^{2} \)
13 \( 1 + (0.253 - 1.59i)T + (-0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (1.34 + 1.34i)T + iT^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.533 + 1.04i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (1.04 + 0.533i)T + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (-1.87 - 0.297i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + 0.618iT - T^{2} \)
97 \( 1 + (0.951 + 0.309i)T^{2} \)
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

−9.549653707153808490620184534899, −8.921037847018156874970792155418, −8.371353824943451038445479154247, −7.21962393932932294771676386571, −6.65901176785158235854850520087, −6.12768195271189450371287676543, −4.77391967477361220680356376470, −3.77166936611060603751599777093, −2.63338689397956863476356084961, −1.90587965956682774652220841075, 0.009145763613288333195028752461, 1.68202583001520694720614296506, 2.91950866748097210970821658056, 3.54987524781502034339372301144, 5.34302591874225257445445431887, 5.70220755327092894117093849917, 6.58061587461407165792272692900, 7.68158637959687232461492580948, 8.096561395481540875032170360923, 8.611887261568878419632218182983

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