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Properties

Label 2-2852-2852.1195-c0-0-4
Degree $2$
Conductor $2852$
Sign $0.957 + 0.289i$
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  + (−0.173 + 0.984i)2-s + (0.748 − 0.831i)3-s + (−0.939 − 0.342i)4-s + (0.688 + 0.881i)6-s + (0.5 − 0.866i)8-s + (−0.0262 − 0.249i)9-s + (−0.987 + 0.525i)12-s + (0.288 − 1.35i)13-s + (0.766 + 0.642i)16-s + (0.250 + 0.0174i)18-s + (0.809 − 0.587i)23-s + (−0.345 − 1.06i)24-s + (−0.5 − 0.866i)25-s + (1.28 + 0.520i)26-s + (0.677 + 0.492i)27-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.748 − 0.831i)3-s + (−0.939 − 0.342i)4-s + (0.688 + 0.881i)6-s + (0.5 − 0.866i)8-s + (−0.0262 − 0.249i)9-s + (−0.987 + 0.525i)12-s + (0.288 − 1.35i)13-s + (0.766 + 0.642i)16-s + (0.250 + 0.0174i)18-s + (0.809 − 0.587i)23-s + (−0.345 − 1.06i)24-s + (−0.5 − 0.866i)25-s + (1.28 + 0.520i)26-s + (0.677 + 0.492i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2852\)    =    \(2^{2} \cdot 23 \cdot 31\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(1.42333\)
Root analytic conductor: \(1.19303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2852} (1195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2852,\ (\ :0),\ 0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.304059767\)
\(L(\frac12)\) \(\approx\) \(1.304059767\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.882 - 0.469i)T \)
good3 \( 1 + (-0.748 + 0.831i)T + (-0.104 - 0.994i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.978 - 0.207i)T^{2} \)
11 \( 1 + (-0.669 + 0.743i)T^{2} \)
13 \( 1 + (-0.288 + 1.35i)T + (-0.913 - 0.406i)T^{2} \)
17 \( 1 + (0.669 + 0.743i)T^{2} \)
19 \( 1 + (0.913 - 0.406i)T^{2} \)
29 \( 1 + (1.84 + 0.599i)T + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.586 + 0.651i)T + (-0.104 + 0.994i)T^{2} \)
43 \( 1 + (-0.913 + 0.406i)T^{2} \)
47 \( 1 + (-0.892 + 0.290i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.978 + 0.207i)T^{2} \)
59 \( 1 + (-0.873 - 0.786i)T + (0.104 + 0.994i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.138 - 0.0145i)T + (0.978 - 0.207i)T^{2} \)
73 \( 1 + (0.224 + 0.503i)T + (-0.669 + 0.743i)T^{2} \)
79 \( 1 + (-0.669 - 0.743i)T^{2} \)
83 \( 1 + (0.104 - 0.994i)T^{2} \)
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

−8.572119721547766642309198383267, −8.151672803666233018011259600419, −7.44772681858734926492688583870, −6.92598188933673252760367279493, −5.95125463623853122772610282282, −5.34554583982136332498906156501, −4.29081900131939172010053731622, −3.27880731497358819287358144501, −2.22868313226346430711893785438, −0.887188863456246225418045694771, 1.46280725321306707241101050389, 2.47510725293451823251482880359, 3.49998459601204119068874453790, 3.93864207361728087917602314053, 4.75948154232665230068241354135, 5.67027066679581651773066643840, 6.90770666124120149370808129921, 7.74881687649458728185818792246, 8.708489275157508424343197334633, 9.143047300227264088063875360346

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