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Properties

Label 2-2888-2888.2747-c0-0-0
Degree $2$
Conductor $2888$
Sign $-0.939 - 0.343i$
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  + (−0.998 + 0.0550i)2-s + (0.289 − 1.48i)3-s + (0.993 − 0.110i)4-s + (−0.207 + 1.49i)6-s + (−0.986 + 0.164i)8-s + (−1.18 − 0.479i)9-s + (−0.934 − 0.727i)11-s + (0.124 − 1.50i)12-s + (0.975 − 0.218i)16-s + (−1.34 − 0.149i)17-s + (1.20 + 0.413i)18-s + (−0.998 − 0.0550i)19-s + (0.973 + 0.674i)22-s + (−0.0415 + 1.50i)24-s + (0.904 − 0.426i)25-s + ⋯
L(s)  = 1  + (−0.998 + 0.0550i)2-s + (0.289 − 1.48i)3-s + (0.993 − 0.110i)4-s + (−0.207 + 1.49i)6-s + (−0.986 + 0.164i)8-s + (−1.18 − 0.479i)9-s + (−0.934 − 0.727i)11-s + (0.124 − 1.50i)12-s + (0.975 − 0.218i)16-s + (−1.34 − 0.149i)17-s + (1.20 + 0.413i)18-s + (−0.998 − 0.0550i)19-s + (0.973 + 0.674i)22-s + (−0.0415 + 1.50i)24-s + (0.904 − 0.426i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.939 - 0.343i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2747, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.939 - 0.343i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4049130672\)
\(L(\frac12)\) \(\approx\) \(0.4049130672\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.998 - 0.0550i)T \)
19 \( 1 + (0.998 + 0.0550i)T \)
good3 \( 1 + (-0.289 + 1.48i)T + (-0.926 - 0.376i)T^{2} \)
5 \( 1 + (-0.904 + 0.426i)T^{2} \)
7 \( 1 + (0.677 - 0.735i)T^{2} \)
11 \( 1 + (0.934 + 0.727i)T + (0.245 + 0.969i)T^{2} \)
13 \( 1 + (-0.137 - 0.990i)T^{2} \)
17 \( 1 + (1.34 + 0.149i)T + (0.975 + 0.218i)T^{2} \)
23 \( 1 + (0.926 - 0.376i)T^{2} \)
29 \( 1 + (0.298 + 0.954i)T^{2} \)
31 \( 1 + (0.401 + 0.915i)T^{2} \)
37 \( 1 + (-0.245 - 0.969i)T^{2} \)
41 \( 1 + (1.28 - 0.789i)T + (0.451 - 0.892i)T^{2} \)
43 \( 1 + (1.05 + 0.297i)T + (0.851 + 0.523i)T^{2} \)
47 \( 1 + (0.962 + 0.272i)T^{2} \)
53 \( 1 + (0.962 + 0.272i)T^{2} \)
59 \( 1 + (-0.769 + 0.472i)T + (0.451 - 0.892i)T^{2} \)
61 \( 1 + (0.754 - 0.656i)T^{2} \)
67 \( 1 + (1.17 + 1.43i)T + (-0.191 + 0.981i)T^{2} \)
71 \( 1 + (0.754 + 0.656i)T^{2} \)
73 \( 1 + (-1.69 - 0.187i)T + (0.975 + 0.218i)T^{2} \)
79 \( 1 + (-0.851 - 0.523i)T^{2} \)
83 \( 1 + (-0.404 + 0.439i)T + (-0.0825 - 0.996i)T^{2} \)
89 \( 1 + (1.34 - 0.149i)T + (0.975 - 0.218i)T^{2} \)
97 \( 1 + (0.635 - 0.771i)T + (-0.191 - 0.981i)T^{2} \)
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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

−8.270014804772915309802868964118, −8.092147815862714776386874982897, −6.95311514457664736619530702062, −6.67606770803030697131205858003, −5.91889322998794672043556397302, −4.78725603607539186432704127943, −3.17099255047698117226107016403, −2.42946695155086106148517015134, −1.65060769140559688124674236777, −0.30847794939142019715992796533, 1.96303343056797952641916462611, 2.81102370938131640792623135495, 3.77769908836341329676832627607, 4.68017596124430829198380158802, 5.37604606843749518072542532071, 6.56865697204074891072848553521, 7.17402926732852416238795935987, 8.364319453044157280468652012122, 8.609715719010905846745276951426, 9.431256460746714523516612645740

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