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Properties

Label 2-2872-2872.717-c0-0-8
Degree $2$
Conductor $2872$
Sign $0.546 - 0.837i$
Analytic cond. $1.43331$
Root an. cond. $1.19721$
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  + (−0.945 + 0.324i)2-s + 1.99i·3-s + (0.789 − 0.614i)4-s − 0.951i·5-s + (−0.647 − 1.88i)6-s + (−0.546 + 0.837i)8-s − 2.97·9-s + (0.309 + 0.900i)10-s − 1.47i·11-s + (1.22 + 1.57i)12-s + 1.89·15-s + (0.245 − 0.969i)16-s − 0.490·17-s + (2.81 − 0.965i)18-s + (−0.584 − 0.751i)20-s + ⋯
L(s)  = 1  + (−0.945 + 0.324i)2-s + 1.99i·3-s + (0.789 − 0.614i)4-s − 0.951i·5-s + (−0.647 − 1.88i)6-s + (−0.546 + 0.837i)8-s − 2.97·9-s + (0.309 + 0.900i)10-s − 1.47i·11-s + (1.22 + 1.57i)12-s + 1.89·15-s + (0.245 − 0.969i)16-s − 0.490·17-s + (2.81 − 0.965i)18-s + (−0.584 − 0.751i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2872\)    =    \(2^{3} \cdot 359\)
Sign: $0.546 - 0.837i$
Analytic conductor: \(1.43331\)
Root analytic conductor: \(1.19721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2872} (717, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2872,\ (\ :0),\ 0.546 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7200314787\)
\(L(\frac12)\) \(\approx\) \(0.7200314787\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.945 - 0.324i)T \)
359 \( 1 + T \)
good3 \( 1 - 1.99iT - T^{2} \)
5 \( 1 + 0.951iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.47iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 0.490T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.57T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.83iT - T^{2} \)
41 \( 1 + 0.803T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.97T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.89T + T^{2} \)
79 \( 1 - 1.75T + T^{2} \)
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

−9.071198477233664593622843943894, −8.580584364089823301099115281323, −8.169932250695185348684292753052, −6.73156819436122813158227303316, −5.79316016883061700463961936961, −5.23254949134704190917343585325, −4.58894316398417814303086656965, −3.46767666569168414532246362857, −2.70462659139407563375256980677, −0.789291251035774917778280643853, 0.996926147869780284629223659688, 2.24685265968568588675226232508, 2.43541930520479999021122715954, 3.59649088817820251541982216035, 5.27743071716837426771516681267, 6.40440962110849833692744327051, 6.86080424519821125356795672387, 7.35367546667088262870367800176, 7.77222476362228758961797980852, 8.865712376729751765292643813110

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