↑

Properties

Label 2-2852-2852.827-c0-0-5
Degree $2$
Conductor $2852$
Sign $0.306 - 0.951i$
Analytic cond. $1.42333$
Root an. cond. $1.19303$
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  + (0.939 + 0.342i)2-s + (0.0783 + 0.745i)3-s + (0.766 + 0.642i)4-s + (−0.181 + 0.726i)6-s + (0.500 + 0.866i)8-s + (0.429 − 0.0912i)9-s + (−0.418 + 0.621i)12-s + (0.812 − 1.82i)13-s + (0.173 + 0.984i)16-s + (0.434 + 0.0610i)18-s + (−0.309 + 0.951i)23-s + (−0.606 + 0.440i)24-s + (−0.5 + 0.866i)25-s + (1.38 − 1.43i)26-s + (0.333 + 1.02i)27-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.0783 + 0.745i)3-s + (0.766 + 0.642i)4-s + (−0.181 + 0.726i)6-s + (0.500 + 0.866i)8-s + (0.429 − 0.0912i)9-s + (−0.418 + 0.621i)12-s + (0.812 − 1.82i)13-s + (0.173 + 0.984i)16-s + (0.434 + 0.0610i)18-s + (−0.309 + 0.951i)23-s + (−0.606 + 0.440i)24-s + (−0.5 + 0.866i)25-s + (1.38 − 1.43i)26-s + (0.333 + 1.02i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2852\)    =    \(2^{2} \cdot 23 \cdot 31\)
Sign: $0.306 - 0.951i$
Analytic conductor: \(1.42333\)
Root analytic conductor: \(1.19303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2852} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2852,\ (\ :0),\ 0.306 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.455692734\)
\(L(\frac12)\) \(\approx\) \(2.455692734\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.559 + 0.829i)T \)
good3 \( 1 + (-0.0783 - 0.745i)T + (-0.978 + 0.207i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.913 + 0.406i)T^{2} \)
11 \( 1 + (0.104 + 0.994i)T^{2} \)
13 \( 1 + (-0.812 + 1.82i)T + (-0.669 - 0.743i)T^{2} \)
17 \( 1 + (-0.104 + 0.994i)T^{2} \)
19 \( 1 + (0.669 - 0.743i)T^{2} \)
29 \( 1 + (-0.551 + 0.759i)T + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.128 - 1.22i)T + (-0.978 - 0.207i)T^{2} \)
43 \( 1 + (-0.669 + 0.743i)T^{2} \)
47 \( 1 + (0.974 + 1.34i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.913 - 0.406i)T^{2} \)
59 \( 1 + (1.89 - 0.198i)T + (0.978 - 0.207i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.0578 + 0.272i)T + (-0.913 + 0.406i)T^{2} \)
73 \( 1 + (0.787 + 0.709i)T + (0.104 + 0.994i)T^{2} \)
79 \( 1 + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (0.978 + 0.207i)T^{2} \)
89 \( 1 + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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

−9.136884956743153724671070877296, −7.999434037946206208832432332235, −7.76987316065470574304142575656, −6.66436112723758883842646489749, −5.83855971374807905852859751555, −5.27011648177042249916440893828, −4.41434214193053383591346186102, −3.54325945939605576034098234260, −3.10805269262124975316865054742, −1.63150877864876275348783100542, 1.41969404461771078948983339505, 2.01626644296740777682886762505, 3.14136339390956421852752849675, 4.24981782634810690637775284197, 4.61696887653936566219435843599, 5.88260279571330873919093322735, 6.60818678077909573922478591698, 6.91828888344072161793326815224, 7.904349648343948220245256002287, 8.795213011201175267939327542992

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