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Properties

Label 2-2883-93.80-c0-0-4
Degree $2$
Conductor $2883$
Sign $0.339 + 0.940i$
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  + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (0.413 + 0.459i)7-s + (0.669 − 0.743i)9-s + (−0.104 − 0.994i)12-s + (0.169 − 1.60i)13-s + (−0.809 − 0.587i)16-s + (0.169 + 1.60i)19-s + (0.564 + 0.251i)21-s + (−0.5 + 0.866i)25-s + (0.309 − 0.951i)27-s + (0.564 − 0.251i)28-s + (−0.499 − 0.866i)36-s + (−0.309 + 0.535i)37-s + (−0.5 − 1.53i)39-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (0.413 + 0.459i)7-s + (0.669 − 0.743i)9-s + (−0.104 − 0.994i)12-s + (0.169 − 1.60i)13-s + (−0.809 − 0.587i)16-s + (0.169 + 1.60i)19-s + (0.564 + 0.251i)21-s + (−0.5 + 0.866i)25-s + (0.309 − 0.951i)27-s + (0.564 − 0.251i)28-s + (−0.499 − 0.866i)36-s + (−0.309 + 0.535i)37-s + (−0.5 − 1.53i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2883\)    =    \(3 \cdot 31^{2}\)
Sign: $0.339 + 0.940i$
Analytic conductor: \(1.43880\)
Root analytic conductor: \(1.19950\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2883} (1196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2883,\ (\ :0),\ 0.339 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.914373029\)
\(L(\frac12)\) \(\approx\) \(1.914373029\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.413 - 0.459i)T + (-0.104 + 0.994i)T^{2} \)
11 \( 1 + (-0.913 + 0.406i)T^{2} \)
13 \( 1 + (-0.169 + 1.60i)T + (-0.978 - 0.207i)T^{2} \)
17 \( 1 + (-0.913 - 0.406i)T^{2} \)
19 \( 1 + (-0.169 - 1.60i)T + (-0.978 + 0.207i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.669 - 0.743i)T^{2} \)
43 \( 1 + (0.0646 + 0.614i)T + (-0.978 + 0.207i)T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.104 + 0.994i)T^{2} \)
59 \( 1 + (-0.669 + 0.743i)T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.104 + 0.994i)T^{2} \)
73 \( 1 + (0.604 - 0.128i)T + (0.913 - 0.406i)T^{2} \)
79 \( 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)T^{2} \)
83 \( 1 + (-0.669 - 0.743i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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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

−8.724222424182734684035372431851, −8.022004041106705131974301895427, −7.49386953539015284830735687053, −6.50631130751540728547635232834, −5.70677171164342058156016413131, −5.18839355372652412170665486592, −3.86745627677336326793415081990, −3.01397807128117815133600408196, −2.02247700774678763511361339053, −1.19815643797045460919526349130, 1.79137696817878890039246410155, 2.59729656692198338127032861611, 3.52398707663446624881369444127, 4.37180322547712534407378339131, 4.73570541465990352333986139534, 6.34516102215481872890641550279, 7.06545719618348471094041181374, 7.64005187684158403248762052516, 8.364271064505546388102922920440, 9.093173215552567915232190532204

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