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Properties

Label 2-2888-2888.235-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.683 - 0.729i$
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  + (0.716 − 0.697i)2-s + (−1.48 + 0.701i)3-s + (0.0275 − 0.999i)4-s + (−0.576 + 1.53i)6-s + (−0.677 − 0.735i)8-s + (1.08 − 1.31i)9-s + (−1.68 + 0.280i)11-s + (0.660 + 1.50i)12-s + (−0.998 − 0.0550i)16-s + (0.0301 + 1.09i)17-s + (−0.140 − 1.69i)18-s + (0.716 + 0.697i)19-s + (−1.00 + 1.37i)22-s + (1.52 + 0.618i)24-s + (0.993 + 0.110i)25-s + ⋯
L(s)  = 1  + (0.716 − 0.697i)2-s + (−1.48 + 0.701i)3-s + (0.0275 − 0.999i)4-s + (−0.576 + 1.53i)6-s + (−0.677 − 0.735i)8-s + (1.08 − 1.31i)9-s + (−1.68 + 0.280i)11-s + (0.660 + 1.50i)12-s + (−0.998 − 0.0550i)16-s + (0.0301 + 1.09i)17-s + (−0.140 − 1.69i)18-s + (0.716 + 0.697i)19-s + (−1.00 + 1.37i)22-s + (1.52 + 0.618i)24-s + (0.993 + 0.110i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.683 - 0.729i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.683 - 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7176192321\)
\(L(\frac12)\) \(\approx\) \(0.7176192321\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.716 + 0.697i)T \)
19 \( 1 + (-0.716 - 0.697i)T \)
good3 \( 1 + (1.48 - 0.701i)T + (0.635 - 0.771i)T^{2} \)
5 \( 1 + (-0.993 - 0.110i)T^{2} \)
7 \( 1 + (-0.546 - 0.837i)T^{2} \)
11 \( 1 + (1.68 - 0.280i)T + (0.945 - 0.324i)T^{2} \)
13 \( 1 + (-0.350 - 0.936i)T^{2} \)
17 \( 1 + (-0.0301 - 1.09i)T + (-0.998 + 0.0550i)T^{2} \)
23 \( 1 + (-0.635 - 0.771i)T^{2} \)
29 \( 1 + (-0.451 + 0.892i)T^{2} \)
31 \( 1 + (0.879 + 0.475i)T^{2} \)
37 \( 1 + (-0.945 + 0.324i)T^{2} \)
41 \( 1 + (0.225 - 1.62i)T + (-0.962 - 0.272i)T^{2} \)
43 \( 1 + (0.370 + 0.322i)T + (0.137 + 0.990i)T^{2} \)
47 \( 1 + (0.754 + 0.656i)T^{2} \)
53 \( 1 + (0.754 + 0.656i)T^{2} \)
59 \( 1 + (0.264 - 1.90i)T + (-0.962 - 0.272i)T^{2} \)
61 \( 1 + (0.821 - 0.569i)T^{2} \)
67 \( 1 + (-1.24 + 0.278i)T + (0.904 - 0.426i)T^{2} \)
71 \( 1 + (0.821 + 0.569i)T^{2} \)
73 \( 1 + (-0.00756 - 0.274i)T + (-0.998 + 0.0550i)T^{2} \)
79 \( 1 + (-0.137 - 0.990i)T^{2} \)
83 \( 1 + (-0.493 - 0.756i)T + (-0.401 + 0.915i)T^{2} \)
89 \( 1 + (-0.0301 + 1.09i)T + (-0.998 - 0.0550i)T^{2} \)
97 \( 1 + (0.975 + 0.218i)T + (0.904 + 0.426i)T^{2} \)
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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

−9.536760955311879408893852229127, −8.357186920053158990914099724296, −7.32269848297527499948733941504, −6.32454748439052040935778311487, −5.72507773461132145885742532528, −5.11717446940769215718107734475, −4.56176693602410897613742610976, −3.63896520975623582681107571917, −2.62534498437381759415690299010, −1.21964868687913879339924870166, 0.46625033313595384492488502218, 2.32036792696074703902074771775, 3.25108700591916356533530820992, 4.69126158427001940423115508557, 5.27108960597096663116707926398, 5.54767094176664819913903303534, 6.64637764645750208245842964268, 7.07648000290660528500592688322, 7.73014155004339030003161474049, 8.512289088981941883711583138572

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