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Properties

Label 2-2888-2888.2859-c0-0-0
Degree $2$
Conductor $2888$
Sign $-0.994 + 0.103i$
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  + (−0.155 + 0.987i)2-s + (0.467 + 1.59i)3-s + (−0.951 − 0.307i)4-s + (−1.65 + 0.213i)6-s + (0.451 − 0.892i)8-s + (−1.49 + 0.954i)9-s + (1.95 + 0.437i)11-s + (0.0458 − 1.66i)12-s + (0.811 + 0.584i)16-s + (1.06 + 0.961i)17-s + (−0.710 − 1.62i)18-s + (−0.777 + 0.628i)19-s + (−0.735 + 1.85i)22-s + (1.63 + 0.304i)24-s + (−0.979 − 0.200i)25-s + ⋯
L(s)  = 1  + (−0.155 + 0.987i)2-s + (0.467 + 1.59i)3-s + (−0.951 − 0.307i)4-s + (−1.65 + 0.213i)6-s + (0.451 − 0.892i)8-s + (−1.49 + 0.954i)9-s + (1.95 + 0.437i)11-s + (0.0458 − 1.66i)12-s + (0.811 + 0.584i)16-s + (1.06 + 0.961i)17-s + (−0.710 − 1.62i)18-s + (−0.777 + 0.628i)19-s + (−0.735 + 1.85i)22-s + (1.63 + 0.304i)24-s + (−0.979 − 0.200i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.994 + 0.103i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2859, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.994 + 0.103i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.291734068\)
\(L(\frac12)\) \(\approx\) \(1.291734068\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.155 - 0.987i)T \)
19 \( 1 + (0.777 - 0.628i)T \)
good3 \( 1 + (-0.467 - 1.59i)T + (-0.842 + 0.539i)T^{2} \)
5 \( 1 + (0.979 + 0.200i)T^{2} \)
7 \( 1 + (0.962 - 0.272i)T^{2} \)
11 \( 1 + (-1.95 - 0.437i)T + (0.904 + 0.426i)T^{2} \)
13 \( 1 + (-0.606 + 0.794i)T^{2} \)
17 \( 1 + (-1.06 - 0.961i)T + (0.100 + 0.994i)T^{2} \)
23 \( 1 + (0.0459 - 0.998i)T^{2} \)
29 \( 1 + (0.562 - 0.826i)T^{2} \)
31 \( 1 + (0.926 + 0.376i)T^{2} \)
37 \( 1 + (0.0825 + 0.996i)T^{2} \)
41 \( 1 + (0.110 - 0.0652i)T + (0.484 - 0.875i)T^{2} \)
43 \( 1 + (-0.397 + 1.45i)T + (-0.861 - 0.507i)T^{2} \)
47 \( 1 + (0.263 - 0.964i)T^{2} \)
53 \( 1 + (0.703 + 0.710i)T^{2} \)
59 \( 1 + (0.00947 + 1.03i)T + (-0.999 + 0.0183i)T^{2} \)
61 \( 1 + (0.896 + 0.443i)T^{2} \)
67 \( 1 + (-1.63 + 0.698i)T + (0.690 - 0.723i)T^{2} \)
71 \( 1 + (0.896 - 0.443i)T^{2} \)
73 \( 1 + (0.361 - 1.68i)T + (-0.912 - 0.410i)T^{2} \)
79 \( 1 + (0.00918 - 0.999i)T^{2} \)
83 \( 1 + (0.806 + 0.784i)T + (0.0275 + 0.999i)T^{2} \)
89 \( 1 + (0.403 + 1.88i)T + (-0.912 + 0.410i)T^{2} \)
97 \( 1 + (1.72 + 0.739i)T + (0.690 + 0.723i)T^{2} \)
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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

−9.364860539020727123067115862190, −8.565849495317119409566198587999, −8.151438604850175781909861005392, −7.05419752318806342602030280548, −6.18399501737839991370505232254, −5.56067726402199515585727102717, −4.53001753611975118219270302395, −3.92926759430471262932591015366, −3.53542817047927540536271992724, −1.68623126758913308006753891750, 0.922782209582470724126666697718, 1.63091432779779704225468912456, 2.63869377462415567011903238000, 3.44313184733751189881798884017, 4.33371300475276794664027684316, 5.60572884423757802002759076336, 6.46904459695792391600096971358, 7.14219885548716256192539854231, 7.980275158974145173181462560756, 8.539334922659082635147764499671

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