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Properties

Label 2-2852-2852.735-c0-0-0
Degree $2$
Conductor $2852$
Sign $0.542 - 0.840i$
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 + (−1.88 − 0.399i)3-s + (−0.939 + 0.342i)4-s + (−0.0670 + 1.92i)6-s + (0.5 + 0.866i)8-s + (2.46 + 1.09i)9-s + (1.90 − 0.267i)12-s + (−1.33 − 1.20i)13-s + (0.766 − 0.642i)16-s + (0.652 − 2.61i)18-s + (0.809 − 0.587i)23-s + (−0.594 − 1.82i)24-s + (−0.5 + 0.866i)25-s + (−0.952 + 1.52i)26-s + (−2.63 − 1.91i)27-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−1.88 − 0.399i)3-s + (−0.939 + 0.342i)4-s + (−0.0670 + 1.92i)6-s + (0.5 + 0.866i)8-s + (2.46 + 1.09i)9-s + (1.90 − 0.267i)12-s + (−1.33 − 1.20i)13-s + (0.766 − 0.642i)16-s + (0.652 − 2.61i)18-s + (0.809 − 0.587i)23-s + (−0.594 − 1.82i)24-s + (−0.5 + 0.866i)25-s + (−0.952 + 1.52i)26-s + (−2.63 − 1.91i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2852\)    =    \(2^{2} \cdot 23 \cdot 31\)
Sign: $0.542 - 0.840i$
Analytic conductor: \(1.42333\)
Root analytic conductor: \(1.19303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2852} (735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2852,\ (\ :0),\ 0.542 - 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09850796891\)
\(L(\frac12)\) \(\approx\) \(0.09850796891\)
\(L(1)\) 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 \( 1 + (0.990 + 0.139i)T \)
good3 \( 1 + (1.88 + 0.399i)T + (0.913 + 0.406i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.669 - 0.743i)T^{2} \)
11 \( 1 + (0.978 + 0.207i)T^{2} \)
13 \( 1 + (1.33 + 1.20i)T + (0.104 + 0.994i)T^{2} \)
17 \( 1 + (-0.978 + 0.207i)T^{2} \)
19 \( 1 + (-0.104 + 0.994i)T^{2} \)
29 \( 1 + (-0.132 - 0.0431i)T + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.40 - 0.299i)T + (0.913 - 0.406i)T^{2} \)
43 \( 1 + (0.104 - 0.994i)T^{2} \)
47 \( 1 + (0.264 - 0.0860i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.669 + 0.743i)T^{2} \)
59 \( 1 + (-0.244 + 1.14i)T + (-0.913 - 0.406i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.789 - 1.77i)T + (-0.669 - 0.743i)T^{2} \)
73 \( 1 + (-1.64 - 0.173i)T + (0.978 + 0.207i)T^{2} \)
79 \( 1 + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (-0.913 + 0.406i)T^{2} \)
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

−9.555732022099706429319531798883, −8.256459449034096542801018375859, −7.48439392513024877426199370106, −6.86191428381017498545671660880, −5.74550306139998292092207586142, −5.13819987791560601751107521811, −4.67558851572843667271856341827, −3.45579753149004663711400517766, −2.22331638249231050848818757886, −1.10297769743596968922672764661, 0.10145357027634961714458102537, 1.65572054567380649916804384402, 3.75799791923324545588337332786, 4.61107241947383280514591073242, 5.07593961824538042093946417635, 5.75433212221252698504484790193, 6.69157320527854570810877023541, 6.93297721463068808735380868597, 7.78345261950497836546025312492, 9.040171270957121160184591873369

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