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Properties

Label 2-2888-2888.283-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.764 + 0.644i$
Analytic cond. $1.44129$
Root an. cond. $1.20054$
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  + (0.919 − 0.393i)2-s + (0.903 + 0.347i)3-s + (0.690 − 0.723i)4-s + (0.967 − 0.0355i)6-s + (0.350 − 0.936i)8-s + (−0.0467 − 0.0422i)9-s + (1.66 − 0.674i)11-s + (0.875 − 0.413i)12-s + (−0.0459 − 0.998i)16-s + (−1.89 + 0.461i)17-s + (−0.0595 − 0.0204i)18-s + (−0.800 + 0.599i)19-s + (1.26 − 1.27i)22-s + (0.642 − 0.724i)24-s + (0.577 + 0.816i)25-s + ⋯
L(s)  = 1  + (0.919 − 0.393i)2-s + (0.903 + 0.347i)3-s + (0.690 − 0.723i)4-s + (0.967 − 0.0355i)6-s + (0.350 − 0.936i)8-s + (−0.0467 − 0.0422i)9-s + (1.66 − 0.674i)11-s + (0.875 − 0.413i)12-s + (−0.0459 − 0.998i)16-s + (−1.89 + 0.461i)17-s + (−0.0595 − 0.0204i)18-s + (−0.800 + 0.599i)19-s + (1.26 − 1.27i)22-s + (0.642 − 0.724i)24-s + (0.577 + 0.816i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.764 + 0.644i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.764 + 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.910854917\)
\(L(\frac12)\) \(\approx\) \(2.910854917\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.919 + 0.393i)T \)
19 \( 1 + (0.800 - 0.599i)T \)
good3 \( 1 + (-0.903 - 0.347i)T + (0.741 + 0.670i)T^{2} \)
5 \( 1 + (-0.577 - 0.816i)T^{2} \)
7 \( 1 + (0.298 + 0.954i)T^{2} \)
11 \( 1 + (-1.66 + 0.674i)T + (0.716 - 0.697i)T^{2} \)
13 \( 1 + (0.467 + 0.883i)T^{2} \)
17 \( 1 + (1.89 - 0.461i)T + (0.888 - 0.459i)T^{2} \)
23 \( 1 + (-0.209 - 0.977i)T^{2} \)
29 \( 1 + (-0.384 + 0.922i)T^{2} \)
31 \( 1 + (-0.993 - 0.110i)T^{2} \)
37 \( 1 + (-0.245 - 0.969i)T^{2} \)
41 \( 1 + (-0.735 - 1.85i)T + (-0.729 + 0.684i)T^{2} \)
43 \( 1 + (0.507 + 0.746i)T + (-0.367 + 0.929i)T^{2} \)
47 \( 1 + (0.562 + 0.826i)T^{2} \)
53 \( 1 + (0.435 - 0.900i)T^{2} \)
59 \( 1 + (1.18 - 1.49i)T + (-0.227 - 0.973i)T^{2} \)
61 \( 1 + (-0.515 - 0.856i)T^{2} \)
67 \( 1 + (-0.634 - 1.79i)T + (-0.777 + 0.628i)T^{2} \)
71 \( 1 + (-0.515 + 0.856i)T^{2} \)
73 \( 1 + (0.206 - 0.705i)T + (-0.842 - 0.539i)T^{2} \)
79 \( 1 + (0.621 + 0.783i)T^{2} \)
83 \( 1 + (-0.750 - 0.168i)T + (0.904 + 0.426i)T^{2} \)
89 \( 1 + (0.167 + 0.572i)T + (-0.842 + 0.539i)T^{2} \)
97 \( 1 + (-0.626 + 1.77i)T + (-0.777 - 0.628i)T^{2} \)
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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

−8.848413015125998220971932620518, −8.446536228191726089138900243324, −7.12677911092849919585989619260, −6.40060526274594301041135818903, −5.91982740686866131557390337298, −4.58555343614932723079889080878, −4.02161276119233505739930703265, −3.39222074194667043579373845905, −2.45766158208634258896765559072, −1.45432919135104983634186339253, 1.90287415218417489301946136327, 2.46106715278416358566378629482, 3.52916728603359304681517269091, 4.37925989799713299722273483404, 4.89748058468690388939121807144, 6.34796958393137879415134407134, 6.61562096458130150906214620822, 7.38335225838562647918226967981, 8.209055051473533037404452874913, 9.012589297259142264515243437228

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