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Properties

Label 2-2852-2852.2851-c0-0-1
Degree $2$
Conductor $2852$
Sign $1$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)6-s − 0.999·8-s + (0.499 + 0.866i)12-s + 1.73i·13-s + (−0.5 + 0.866i)16-s − 23-s + 0.999·24-s + 25-s + (1.49 + 0.866i)26-s + 27-s + 1.73i·29-s + (0.5 − 0.866i)31-s + (0.499 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)6-s − 0.999·8-s + (0.499 + 0.866i)12-s + 1.73i·13-s + (−0.5 + 0.866i)16-s − 23-s + 0.999·24-s + 25-s + (1.49 + 0.866i)26-s + 27-s + 1.73i·29-s + (0.5 − 0.866i)31-s + (0.499 + 0.866i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2852\)    =    \(2^{2} \cdot 23 \cdot 31\)
Sign: $1$
Analytic conductor: \(1.42333\)
Root analytic conductor: \(1.19303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2852} (2851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2852,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7928837008\)
\(L(\frac12)\) \(\approx\) \(0.7928837008\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
31 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.73iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.166004330954093218507962975682, −8.480684609932831077614975191634, −7.18864855914829118803300134968, −6.34012642257038211239342359748, −5.91657760772309839509635995201, −4.83355266915703173205550625408, −4.46340038969518500667561365328, −3.37370016279468278877271913833, −2.29285759895093723683484399654, −1.20546104116754402397475278536, 0.54144781734206160651959865021, 2.62726955131290470756069390547, 3.52315404555034068013279857019, 4.60238970677080179669485803675, 5.26721198614980605489460643393, 5.91481430441962168942541128802, 6.41448906207686750788326510697, 7.35178147028840416132126341252, 8.127910662317978835435286436358, 8.603015688444466992249937353574

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