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Properties

Label 2-2883-2883.326-c0-0-0
Degree $2$
Conductor $2883$
Sign $0.770 + 0.637i$
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  + (0.771 − 0.635i)3-s + (0.717 + 0.696i)4-s + (0.620 − 1.09i)7-s + (0.191 − 0.981i)9-s + (0.996 + 0.0809i)12-s + (−1.55 − 0.190i)13-s + (0.0303 + 0.999i)16-s + (1.67 + 0.309i)19-s + (−0.215 − 1.23i)21-s + (−0.994 + 0.101i)25-s + (−0.476 − 0.879i)27-s + (1.20 − 0.351i)28-s + (0.992 + 0.121i)31-s + (0.820 − 0.571i)36-s + (0.954 − 0.530i)37-s + ⋯
L(s)  = 1  + (0.771 − 0.635i)3-s + (0.717 + 0.696i)4-s + (0.620 − 1.09i)7-s + (0.191 − 0.981i)9-s + (0.996 + 0.0809i)12-s + (−1.55 − 0.190i)13-s + (0.0303 + 0.999i)16-s + (1.67 + 0.309i)19-s + (−0.215 − 1.23i)21-s + (−0.994 + 0.101i)25-s + (−0.476 − 0.879i)27-s + (1.20 − 0.351i)28-s + (0.992 + 0.121i)31-s + (0.820 − 0.571i)36-s + (0.954 − 0.530i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2883\)    =    \(3 \cdot 31^{2}\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(1.43880\)
Root analytic conductor: \(1.19950\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2883} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2883,\ (\ :0),\ 0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.966128448\)
\(L(\frac12)\) \(\approx\) \(1.966128448\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.771 + 0.635i)T \)
31 \( 1 + (-0.992 - 0.121i)T \)
good2 \( 1 + (-0.717 - 0.696i)T^{2} \)
5 \( 1 + (0.994 - 0.101i)T^{2} \)
7 \( 1 + (-0.620 + 1.09i)T + (-0.511 - 0.859i)T^{2} \)
11 \( 1 + (-0.595 - 0.803i)T^{2} \)
13 \( 1 + (1.55 + 0.190i)T + (0.970 + 0.240i)T^{2} \)
17 \( 1 + (0.983 + 0.181i)T^{2} \)
19 \( 1 + (-1.67 - 0.309i)T + (0.934 + 0.356i)T^{2} \)
23 \( 1 + (-0.864 + 0.502i)T^{2} \)
29 \( 1 + (0.403 - 0.914i)T^{2} \)
37 \( 1 + (-0.954 + 0.530i)T + (0.528 - 0.848i)T^{2} \)
41 \( 1 + (-0.111 + 0.993i)T^{2} \)
43 \( 1 + (1.76 - 0.0358i)T + (0.999 - 0.0405i)T^{2} \)
47 \( 1 + (0.211 + 0.977i)T^{2} \)
53 \( 1 + (-0.992 + 0.121i)T^{2} \)
59 \( 1 + (-0.970 - 0.240i)T^{2} \)
61 \( 1 + (0.168 - 0.144i)T + (0.151 - 0.988i)T^{2} \)
67 \( 1 + (-1.46 - 0.628i)T + (0.688 + 0.724i)T^{2} \)
71 \( 1 + (-0.628 - 0.778i)T^{2} \)
73 \( 1 + (1.66 - 0.0674i)T + (0.996 - 0.0809i)T^{2} \)
79 \( 1 + (0.952 + 0.0386i)T + (0.996 + 0.0809i)T^{2} \)
83 \( 1 + (-0.717 - 0.696i)T^{2} \)
89 \( 1 + (0.975 + 0.221i)T^{2} \)
97 \( 1 + (1.30 - 0.186i)T + (0.960 - 0.279i)T^{2} \)
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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

−8.558915590059466614135117028688, −7.86837144204024038075228009808, −7.40234255406235004995034507872, −7.08015419556493021705940863103, −6.02288263870123363713273282741, −4.85146966458443197915782210837, −3.91241286955151395332739388518, −3.11132540901690409873278977573, −2.28948236254062512621809135385, −1.24886128903944858423116232369, 1.68454492420480359922569423963, 2.49767383987856688294033756024, 3.10144098243336109310135195447, 4.57008077460063095489920111557, 5.13736673706641443656140269921, 5.75422339988360053220857427580, 6.89300225765110186534343925418, 7.64577273631999843426510076074, 8.248458489463150120317651970384, 9.220715700696761320771021082181

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