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Properties

Label 2-2888-2888.427-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.874 - 0.485i$
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  + (0.606 − 0.794i)2-s + (0.793 + 1.64i)3-s + (−0.263 − 0.964i)4-s + (1.78 + 0.366i)6-s + (−0.926 − 0.376i)8-s + (−1.44 + 1.82i)9-s + (0.983 + 0.604i)11-s + (1.37 − 1.19i)12-s + (−0.861 + 0.507i)16-s + (0.678 − 0.178i)17-s + (0.571 + 2.25i)18-s + (0.384 − 0.922i)19-s + (1.07 − 0.414i)22-s + (−0.117 − 1.82i)24-s + (0.515 + 0.856i)25-s + ⋯
L(s)  = 1  + (0.606 − 0.794i)2-s + (0.793 + 1.64i)3-s + (−0.263 − 0.964i)4-s + (1.78 + 0.366i)6-s + (−0.926 − 0.376i)8-s + (−1.44 + 1.82i)9-s + (0.983 + 0.604i)11-s + (1.37 − 1.19i)12-s + (−0.861 + 0.507i)16-s + (0.678 − 0.178i)17-s + (0.571 + 2.25i)18-s + (0.384 − 0.922i)19-s + (1.07 − 0.414i)22-s + (−0.117 − 1.82i)24-s + (0.515 + 0.856i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.874 - 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.158068702\)
\(L(\frac12)\) \(\approx\) \(2.158068702\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.606 + 0.794i)T \)
19 \( 1 + (-0.384 + 0.922i)T \)
good3 \( 1 + (-0.793 - 1.64i)T + (-0.621 + 0.783i)T^{2} \)
5 \( 1 + (-0.515 - 0.856i)T^{2} \)
7 \( 1 + (-0.635 - 0.771i)T^{2} \)
11 \( 1 + (-0.983 - 0.604i)T + (0.451 + 0.892i)T^{2} \)
13 \( 1 + (-0.315 + 0.948i)T^{2} \)
17 \( 1 + (-0.678 + 0.178i)T + (0.870 - 0.492i)T^{2} \)
23 \( 1 + (0.367 - 0.929i)T^{2} \)
29 \( 1 + (-0.983 - 0.182i)T^{2} \)
31 \( 1 + (0.821 - 0.569i)T^{2} \)
37 \( 1 + (-0.546 + 0.837i)T^{2} \)
41 \( 1 + (-0.0566 - 0.193i)T + (-0.842 + 0.539i)T^{2} \)
43 \( 1 + (1.56 - 1.17i)T + (0.280 - 0.959i)T^{2} \)
47 \( 1 + (0.800 - 0.599i)T^{2} \)
53 \( 1 + (0.119 + 0.992i)T^{2} \)
59 \( 1 + (-0.0892 - 0.0217i)T + (0.888 + 0.459i)T^{2} \)
61 \( 1 + (-0.811 - 0.584i)T^{2} \)
67 \( 1 + (0.925 + 1.74i)T + (-0.562 + 0.826i)T^{2} \)
71 \( 1 + (-0.811 + 0.584i)T^{2} \)
73 \( 1 + (0.395 + 0.399i)T + (-0.00918 + 0.999i)T^{2} \)
79 \( 1 + (0.971 - 0.236i)T^{2} \)
83 \( 1 + (-0.689 + 1.84i)T + (-0.754 - 0.656i)T^{2} \)
89 \( 1 + (0.894 - 0.903i)T + (-0.00918 - 0.999i)T^{2} \)
97 \( 1 + (-0.879 + 1.66i)T + (-0.562 - 0.826i)T^{2} \)
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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

−9.257019682446678233303155171948, −8.773461153849063248218653809006, −7.65300203140279019993477275277, −6.56160218137280126596130020401, −5.48720083665038412506230283011, −4.77371330754555399778323633845, −4.28572006449585193603914702111, −3.33681507372962497058472094035, −2.91723729996764415040742717410, −1.63020741784584209875996534463, 1.15759595601042326267766317679, 2.35338672263975950723748938600, 3.33469528263143336817718318123, 3.91273903885098372264012672033, 5.36478700346449033679232796091, 6.08663404846714388036499238299, 6.70990833503649415954458016889, 7.27041395793717318652772016454, 8.070689718045535363168838764898, 8.519624837167056367745062343208

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