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Properties

Label 2-2883-93.8-c0-0-3
Degree $2$
Conductor $2883$
Sign $0.400 - 0.916i$
Analytic cond. $1.43880$
Root an. cond. $1.19950$
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  + (−0.951 − 0.309i)2-s + (0.891 + 0.453i)3-s + i·5-s + (−0.707 − 0.707i)6-s + (0.809 + 0.587i)7-s + (0.587 + 0.809i)8-s + (0.587 + 0.809i)9-s + (0.309 − 0.951i)10-s + (−0.587 − 0.809i)14-s + (−0.453 + 0.891i)15-s + (−0.309 − 0.951i)16-s + (0.831 + 1.14i)17-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (0.453 + 0.891i)21-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.891 + 0.453i)3-s + i·5-s + (−0.707 − 0.707i)6-s + (0.809 + 0.587i)7-s + (0.587 + 0.809i)8-s + (0.587 + 0.809i)9-s + (0.309 − 0.951i)10-s + (−0.587 − 0.809i)14-s + (−0.453 + 0.891i)15-s + (−0.309 − 0.951i)16-s + (0.831 + 1.14i)17-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (0.453 + 0.891i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2883\)    =    \(3 \cdot 31^{2}\)
Sign: $0.400 - 0.916i$
Analytic conductor: \(1.43880\)
Root analytic conductor: \(1.19950\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2883} (1589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2883,\ (\ :0),\ 0.400 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.092934012\)
\(L(\frac12)\) \(\approx\) \(1.092934012\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.891 - 0.453i)T \)
31 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
5 \( 1 - iT - T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.437 + 1.34i)T + (-0.809 - 0.587i)T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
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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

−9.008579847035293319327927510571, −8.595520352565604644131223032540, −7.65661863552778520661381425646, −7.41953693207237687337818489470, −6.01818963188067452049172593240, −5.16971978703371114802511619488, −4.28243713792461162515116043213, −3.24008069489062473110470943057, −2.35814125417787083687968775725, −1.59237378433996172383324377698, 0.993122028406408756366660616479, 1.59923950656135862714979645379, 3.12939877433493126443514739097, 4.11319415581871945013680678909, 4.77420014963122574265032111687, 5.85358428021019307181421315434, 7.09868819491987306264274135538, 7.67338668065130768717175036639, 7.955938927728773725466386570092, 8.788475397818272521151070462584

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