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Properties

Label 2-2852-2852.275-c0-0-1
Degree $2$
Conductor $2852$
Sign $-0.629 - 0.777i$
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 + (−0.0646 + 0.198i)3-s + (−0.499 + 0.866i)4-s + (−0.204 + 0.0434i)6-s − 0.999·8-s + (0.773 + 0.562i)9-s + (−0.139 − 0.155i)12-s + (0.773 + 0.251i)13-s + (−0.5 − 0.866i)16-s + (−0.0999 + 0.951i)18-s + (0.809 + 0.587i)23-s + (0.0646 − 0.198i)24-s + 25-s + (0.169 + 0.795i)26-s + (−0.330 + 0.240i)27-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.0646 + 0.198i)3-s + (−0.499 + 0.866i)4-s + (−0.204 + 0.0434i)6-s − 0.999·8-s + (0.773 + 0.562i)9-s + (−0.139 − 0.155i)12-s + (0.773 + 0.251i)13-s + (−0.5 − 0.866i)16-s + (−0.0999 + 0.951i)18-s + (0.809 + 0.587i)23-s + (0.0646 − 0.198i)24-s + 25-s + (0.169 + 0.795i)26-s + (−0.330 + 0.240i)27-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2852\)    =    \(2^{2} \cdot 23 \cdot 31\)
Sign: $-0.629 - 0.777i$
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: \((2,\ 2852,\ (\ :0),\ -0.629 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530050155\)
\(L(\frac12)\) \(\approx\) \(1.530050155\)
\(L(1)\) 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 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
good3 \( 1 + (0.0646 - 0.198i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.309 - 0.951i)T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.773 - 0.251i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (0.773 - 0.251i)T + (0.809 - 0.587i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.0646 - 0.198i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (1.41 + 0.459i)T + (0.809 + 0.587i)T^{2} \)
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 \( 1 + (-1.16 + 1.60i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.478 - 0.658i)T + (-0.309 + 0.951i)T^{2} \)
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

−9.121710710195538564767350534578, −8.356407604842604916512451382195, −7.57907448914302510043279497721, −6.94251697897009653566023371777, −6.25599980208561143587340640552, −5.24702270290700234903202863925, −4.77903473730324510729771883142, −3.81815655996130871553738090977, −3.08604469802463628638499750200, −1.59783276273504430030338525005, 0.914112506371285508166954423050, 1.92584874789054380609397856705, 3.09217533817377620208274881474, 3.82443311159714043775227640657, 4.63973544555561031613487561588, 5.49603451734086767327835643548, 6.35141465888085399546847119104, 6.96467001786443147512051165552, 8.053031577570453445971562077252, 8.940567052272493196862341782727

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