Properties

Label 2-4410-21.20-c1-0-41
Degree $2$
Conductor $4410$
Sign $-0.995 + 0.0980i$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
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  i·2-s − 4-s + 5-s + i·8-s i·10-s − 1.53i·11-s + 1.48i·13-s + 16-s − 2.42·17-s − 4.86i·19-s − 20-s − 1.53·22-s − 0.267i·23-s + 25-s + 1.48·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.447·5-s + 0.353i·8-s − 0.316i·10-s − 0.461i·11-s + 0.411i·13-s + 0.250·16-s − 0.589·17-s − 1.11i·19-s − 0.223·20-s − 0.326·22-s − 0.0558i·23-s + 0.200·25-s + 0.290·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.995 + 0.0980i$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4410} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ -0.995 + 0.0980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9905210845\)
\(L(\frac12)\) \(\approx\) \(0.9905210845\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 1.53iT - 11T^{2} \)
13 \( 1 - 1.48iT - 13T^{2} \)
17 \( 1 + 2.42T + 17T^{2} \)
19 \( 1 + 4.86iT - 19T^{2} \)
23 \( 1 + 0.267iT - 23T^{2} \)
29 \( 1 + 0.898iT - 29T^{2} \)
31 \( 1 + 0.828iT - 31T^{2} \)
37 \( 1 + 5.48T + 37T^{2} \)
41 \( 1 + 8.76T + 41T^{2} \)
43 \( 1 - 1.86T + 43T^{2} \)
47 \( 1 - 7.45T + 47T^{2} \)
53 \( 1 - 3.47iT - 53T^{2} \)
59 \( 1 + 6.25T + 59T^{2} \)
61 \( 1 + 6.62iT - 61T^{2} \)
67 \( 1 - 16.0T + 67T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 - 0.343iT - 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 5.45T + 83T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 + 14.9iT - 97T^{2} \)
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

−8.277092272554906134974072531353, −7.19673321368621034115216025122, −6.57260596859457581533476237454, −5.69070553828087357509416933867, −4.92399841377222002066511504385, −4.19062104554582808486983968169, −3.23396974342947320661707904740, −2.43223160127916858898359736240, −1.55252516738445416747450671253, −0.27434825705280080795486463784, 1.32259258796413273323949731623, 2.37894434976663706720384106985, 3.51421301937768711336184799320, 4.30799406560613346115428727016, 5.24748936680021062512816709428, 5.71757599721141004707251628183, 6.63588456047190169157130890201, 7.12388466018307750943200957516, 8.004410145478754028457061378342, 8.589686628356546280904213812343

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