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Properties

Label 2-2883-2883.95-c0-0-0
Degree $2$
Conductor $2883$
Sign $-0.487 + 0.872i$
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  + (0.595 − 0.803i)3-s + (0.385 − 0.922i)4-s + (0.471 − 0.560i)7-s + (−0.289 − 0.957i)9-s + (−0.511 − 0.859i)12-s + (−0.222 + 0.00450i)13-s + (−0.703 − 0.710i)16-s + (−0.00553 − 0.182i)19-s + (−0.169 − 0.712i)21-s + (−0.874 + 0.485i)25-s + (−0.941 − 0.337i)27-s + (−0.335 − 0.650i)28-s + (−0.999 + 0.0202i)31-s + (−0.994 − 0.101i)36-s + (1.26 + 0.881i)37-s + ⋯
L(s)  = 1  + (0.595 − 0.803i)3-s + (0.385 − 0.922i)4-s + (0.471 − 0.560i)7-s + (−0.289 − 0.957i)9-s + (−0.511 − 0.859i)12-s + (−0.222 + 0.00450i)13-s + (−0.703 − 0.710i)16-s + (−0.00553 − 0.182i)19-s + (−0.169 − 0.712i)21-s + (−0.874 + 0.485i)25-s + (−0.941 − 0.337i)27-s + (−0.335 − 0.650i)28-s + (−0.999 + 0.0202i)31-s + (−0.994 − 0.101i)36-s + (1.26 + 0.881i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2883\)    =    \(3 \cdot 31^{2}\)
Sign: $-0.487 + 0.872i$
Analytic conductor: \(1.43880\)
Root analytic conductor: \(1.19950\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2883} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2883,\ (\ :0),\ -0.487 + 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.633075487\)
\(L(\frac12)\) \(\approx\) \(1.633075487\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.595 + 0.803i)T \)
31 \( 1 + (0.999 - 0.0202i)T \)
good2 \( 1 + (-0.385 + 0.922i)T^{2} \)
5 \( 1 + (0.874 - 0.485i)T^{2} \)
7 \( 1 + (-0.471 + 0.560i)T + (-0.171 - 0.985i)T^{2} \)
11 \( 1 + (-0.628 - 0.778i)T^{2} \)
13 \( 1 + (0.222 - 0.00450i)T + (0.999 - 0.0405i)T^{2} \)
17 \( 1 + (-0.0303 - 0.999i)T^{2} \)
19 \( 1 + (0.00553 + 0.182i)T + (-0.998 + 0.0607i)T^{2} \)
23 \( 1 + (-0.422 - 0.906i)T^{2} \)
29 \( 1 + (-0.191 + 0.981i)T^{2} \)
37 \( 1 + (-1.26 - 0.881i)T + (0.347 + 0.937i)T^{2} \)
41 \( 1 + (-0.970 - 0.240i)T^{2} \)
43 \( 1 + (-1.72 - 1.00i)T + (0.494 + 0.869i)T^{2} \)
47 \( 1 + (0.731 - 0.681i)T^{2} \)
53 \( 1 + (0.999 + 0.0202i)T^{2} \)
59 \( 1 + (-0.999 + 0.0405i)T^{2} \)
61 \( 1 + (-1.78 + 0.765i)T + (0.688 - 0.724i)T^{2} \)
67 \( 1 + (-0.355 + 0.724i)T + (-0.612 - 0.790i)T^{2} \)
71 \( 1 + (0.366 - 0.930i)T^{2} \)
73 \( 1 + (0.900 + 1.58i)T + (-0.511 + 0.859i)T^{2} \)
79 \( 1 + (0.930 - 1.63i)T + (-0.511 - 0.859i)T^{2} \)
83 \( 1 + (-0.385 + 0.922i)T^{2} \)
89 \( 1 + (-0.884 - 0.467i)T^{2} \)
97 \( 1 + (-1.69 - 1.03i)T + (0.458 + 0.888i)T^{2} \)
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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

−8.690188601547499777497451065331, −7.66113419224526309402280547802, −7.40160499587209096186543251939, −6.43141747784734068741754591575, −5.85378156518021553403077120519, −4.86736311454242936626073373133, −3.90017544126935526274208044612, −2.74358909038431712635127811794, −1.87763185530038035368623601368, −0.962274415855053774503557786996, 2.10465741023724243036541402344, 2.61283796828999544315454387567, 3.75842707220272478641976445203, 4.21977119610939015171198976597, 5.30320645597771071173587647907, 6.03040076152490236538037226336, 7.32870261330368394686405613236, 7.69481352982110481908040458125, 8.621094423344006766827945573516, 8.933297466442355999138056396369

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