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Properties

Label 2-2888-2888.315-c0-0-0
Degree $2$
Conductor $2888$
Sign $-0.655 + 0.755i$
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  + (0.975 − 0.218i)2-s + (−1.37 − 1.34i)3-s + (0.904 − 0.426i)4-s + (−1.63 − 1.00i)6-s + (0.789 − 0.614i)8-s + (0.0745 + 2.70i)9-s + (1.44 − 0.782i)11-s + (−1.82 − 0.624i)12-s + (0.635 − 0.771i)16-s + (−1.78 − 0.841i)17-s + (0.663 + 2.62i)18-s + (0.975 + 0.218i)19-s + (1.23 − 1.07i)22-s + (−1.91 − 0.211i)24-s + (−0.191 − 0.981i)25-s + ⋯
L(s)  = 1  + (0.975 − 0.218i)2-s + (−1.37 − 1.34i)3-s + (0.904 − 0.426i)4-s + (−1.63 − 1.00i)6-s + (0.789 − 0.614i)8-s + (0.0745 + 2.70i)9-s + (1.44 − 0.782i)11-s + (−1.82 − 0.624i)12-s + (0.635 − 0.771i)16-s + (−1.78 − 0.841i)17-s + (0.663 + 2.62i)18-s + (0.975 + 0.218i)19-s + (1.23 − 1.07i)22-s + (−1.91 − 0.211i)24-s + (−0.191 − 0.981i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.655 + 0.755i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.655 + 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.509362323\)
\(L(\frac12)\) \(\approx\) \(1.509362323\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.975 + 0.218i)T \)
19 \( 1 + (-0.975 - 0.218i)T \)
good3 \( 1 + (1.37 + 1.34i)T + (0.0275 + 0.999i)T^{2} \)
5 \( 1 + (0.191 + 0.981i)T^{2} \)
7 \( 1 + (0.986 - 0.164i)T^{2} \)
11 \( 1 + (-1.44 + 0.782i)T + (0.546 - 0.837i)T^{2} \)
13 \( 1 + (-0.851 + 0.523i)T^{2} \)
17 \( 1 + (1.78 + 0.841i)T + (0.635 + 0.771i)T^{2} \)
23 \( 1 + (-0.0275 + 0.999i)T^{2} \)
29 \( 1 + (-0.350 + 0.936i)T^{2} \)
31 \( 1 + (0.0825 + 0.996i)T^{2} \)
37 \( 1 + (-0.546 + 0.837i)T^{2} \)
41 \( 1 + (-1.13 - 1.55i)T + (-0.298 + 0.954i)T^{2} \)
43 \( 1 + (0.611 + 1.20i)T + (-0.592 + 0.805i)T^{2} \)
47 \( 1 + (-0.451 - 0.892i)T^{2} \)
53 \( 1 + (-0.451 - 0.892i)T^{2} \)
59 \( 1 + (-0.353 - 0.481i)T + (-0.298 + 0.954i)T^{2} \)
61 \( 1 + (0.962 + 0.272i)T^{2} \)
67 \( 1 + (0.0510 + 0.0207i)T + (0.716 + 0.697i)T^{2} \)
71 \( 1 + (0.962 - 0.272i)T^{2} \)
73 \( 1 + (1.07 + 0.505i)T + (0.635 + 0.771i)T^{2} \)
79 \( 1 + (0.592 - 0.805i)T^{2} \)
83 \( 1 + (0.691 - 0.115i)T + (0.945 - 0.324i)T^{2} \)
89 \( 1 + (1.78 - 0.841i)T + (0.635 - 0.771i)T^{2} \)
97 \( 1 + (-0.926 + 0.376i)T + (0.716 - 0.697i)T^{2} \)
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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

−8.464300469904923436979780858688, −7.47463179597628302963362040075, −6.80771401684676786583292696679, −6.35472583220236580549222556944, −5.78694070799036208491387722794, −4.89047038363098538679499305672, −4.19393083849169159458745221484, −2.83530954688828944315696681449, −1.81822319024247425566667516373, −0.869622962839568824502028031169, 1.62769179867024522127051610413, 3.25859476959660894082457519181, 4.16834181495080594076046545687, 4.40485290009035044447499495498, 5.30754601777326017025185798332, 6.01232314836011934672547627562, 6.66992936897290056399835479756, 7.19953504533555793872981899053, 8.687444440212359276032709283199, 9.404502186820727931489426717709

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