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Properties

Label 2-2888-2888.419-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.994 + 0.104i$
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  + (−0.986 − 0.164i)2-s + (0.598 + 0.915i)3-s + (0.945 + 0.324i)4-s + (−0.439 − 1.00i)6-s + (−0.879 − 0.475i)8-s + (−0.0789 + 0.180i)9-s + (−0.759 − 1.73i)11-s + (0.268 + 1.06i)12-s + (0.789 + 0.614i)16-s + (1.49 − 0.512i)17-s + (0.107 − 0.164i)18-s + (−0.986 + 0.164i)19-s + (0.464 + 1.83i)22-s + (−0.0903 − 1.09i)24-s + (0.245 + 0.969i)25-s + ⋯
L(s)  = 1  + (−0.986 − 0.164i)2-s + (0.598 + 0.915i)3-s + (0.945 + 0.324i)4-s + (−0.439 − 1.00i)6-s + (−0.879 − 0.475i)8-s + (−0.0789 + 0.180i)9-s + (−0.759 − 1.73i)11-s + (0.268 + 1.06i)12-s + (0.789 + 0.614i)16-s + (1.49 − 0.512i)17-s + (0.107 − 0.164i)18-s + (−0.986 + 0.164i)19-s + (0.464 + 1.83i)22-s + (−0.0903 − 1.09i)24-s + (0.245 + 0.969i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.994 + 0.104i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.994 + 0.104i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9679654113\)
\(L(\frac12)\) \(\approx\) \(0.9679654113\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.986 + 0.164i)T \)
19 \( 1 + (0.986 - 0.164i)T \)
good3 \( 1 + (-0.598 - 0.915i)T + (-0.401 + 0.915i)T^{2} \)
5 \( 1 + (-0.245 - 0.969i)T^{2} \)
7 \( 1 + (-0.789 + 0.614i)T^{2} \)
11 \( 1 + (0.759 + 1.73i)T + (-0.677 + 0.735i)T^{2} \)
13 \( 1 + (0.401 - 0.915i)T^{2} \)
17 \( 1 + (-1.49 + 0.512i)T + (0.789 - 0.614i)T^{2} \)
23 \( 1 + (0.401 + 0.915i)T^{2} \)
29 \( 1 + (-0.789 + 0.614i)T^{2} \)
31 \( 1 + (-0.945 + 0.324i)T^{2} \)
37 \( 1 + (0.677 - 0.735i)T^{2} \)
41 \( 1 + (0.0903 - 1.09i)T + (-0.986 - 0.164i)T^{2} \)
43 \( 1 + (-1.33 + 1.45i)T + (-0.0825 - 0.996i)T^{2} \)
47 \( 1 + (0.677 - 0.735i)T^{2} \)
53 \( 1 + (0.677 - 0.735i)T^{2} \)
59 \( 1 + (-0.162 + 1.96i)T + (-0.986 - 0.164i)T^{2} \)
61 \( 1 + (-0.546 + 0.837i)T^{2} \)
67 \( 1 + (-0.706 - 0.382i)T + (0.546 + 0.837i)T^{2} \)
71 \( 1 + (-0.546 - 0.837i)T^{2} \)
73 \( 1 + (0.156 - 0.0536i)T + (0.789 - 0.614i)T^{2} \)
79 \( 1 + (0.0825 + 0.996i)T^{2} \)
83 \( 1 + (-1.24 + 0.969i)T + (0.245 - 0.969i)T^{2} \)
89 \( 1 + (-1.49 - 0.512i)T + (0.789 + 0.614i)T^{2} \)
97 \( 1 + (1.75 - 0.951i)T + (0.546 - 0.837i)T^{2} \)
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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

−9.033475742823242637499898150972, −8.283947073532433640155592763201, −7.86766268961066615944214692860, −6.83475346578457287077085214390, −5.89720587515102909805258686510, −5.17897065531106680156368336476, −3.73839864177032295335997009019, −3.30686283926630856563676115777, −2.45243669498125797724853485743, −0.883299972910153887499100364631, 1.25235056837554865843867734351, 2.21598419399882605286730480717, 2.73023488851265804310758239839, 4.22855132038043980397649379700, 5.30375304881016217262206466655, 6.26323157350162927665290004811, 7.03232065549733683965453114331, 7.67292307453883284674201170811, 7.976231922879828525865298453029, 8.824181070469441952792154766765

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