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Properties

Label 2-2852-2852.275-c0-0-3
Degree $2$
Conductor $2852$
Sign $0.777 + 0.629i$
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  + (−0.913 − 0.406i)2-s + (−0.564 + 1.73i)3-s + (0.669 + 0.743i)4-s + (1.22 − 1.35i)6-s + (−0.309 − 0.951i)8-s + (−1.89 − 1.37i)9-s + (−1.66 + 0.743i)12-s + (−1.89 − 0.614i)13-s + (−0.104 + 0.994i)16-s + (1.16 + 2.02i)18-s + (−0.809 − 0.587i)23-s + 1.82·24-s + 25-s + (1.47 + 1.33i)26-s + (1.97 − 1.43i)27-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (−0.564 + 1.73i)3-s + (0.669 + 0.743i)4-s + (1.22 − 1.35i)6-s + (−0.309 − 0.951i)8-s + (−1.89 − 1.37i)9-s + (−1.66 + 0.743i)12-s + (−1.89 − 0.614i)13-s + (−0.104 + 0.994i)16-s + (1.16 + 2.02i)18-s + (−0.809 − 0.587i)23-s + 1.82·24-s + 25-s + (1.47 + 1.33i)26-s + (1.97 − 1.43i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2852\)    =    \(2^{2} \cdot 23 \cdot 31\)
Sign: $0.777 + 0.629i$
Analytic conductor: \(1.42333\)
Root analytic conductor: \(1.19303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2852} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2852,\ (\ :0),\ 0.777 + 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3385495373\)
\(L(\frac12)\) \(\approx\) \(0.3385495373\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
good3 \( 1 + (0.564 - 1.73i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.309 - 0.951i)T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.89 + 0.614i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-1.89 + 0.614i)T + (0.809 - 0.587i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.564 + 1.73i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (-0.395 - 0.128i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.478 + 0.658i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.16 + 1.60i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−9.117869055923404226821050747730, −8.456754932370173808893201378701, −7.54981462715536155814455988211, −6.64516235813014918492392809150, −5.68716499718705595614025669205, −4.83885626808421013839423463312, −4.18401988705167562343664313423, −3.15156434888674989041703722202, −2.42211775081542353070426344053, −0.33897847369975363845915184109, 1.12171670676367907298370015111, 2.07930115790803200310554484639, 2.80753907418367536663377087618, 4.85371319607012869347891004354, 5.43681047565695392550424840414, 6.45104799727964179645410963835, 6.90495843352929095023931888610, 7.38564811659569558457966055038, 8.146097358325632942903091611439, 8.738820933855710927789216544684

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