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Properties

Label 2-2888-2888.2699-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.799 + 0.600i$
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  + (−0.879 − 0.475i)2-s + (1.94 − 0.324i)3-s + (0.546 + 0.837i)4-s + (−1.86 − 0.640i)6-s + (−0.0825 − 0.996i)8-s + (2.73 − 0.938i)9-s + (1.03 + 0.355i)11-s + (1.33 + 1.45i)12-s + (−0.401 + 0.915i)16-s + (−0.439 + 0.672i)17-s + (−2.85 − 0.475i)18-s + (−0.879 + 0.475i)19-s + (−0.740 − 0.804i)22-s + (−0.484 − 1.91i)24-s + (−0.677 − 0.735i)25-s + ⋯
L(s)  = 1  + (−0.879 − 0.475i)2-s + (1.94 − 0.324i)3-s + (0.546 + 0.837i)4-s + (−1.86 − 0.640i)6-s + (−0.0825 − 0.996i)8-s + (2.73 − 0.938i)9-s + (1.03 + 0.355i)11-s + (1.33 + 1.45i)12-s + (−0.401 + 0.915i)16-s + (−0.439 + 0.672i)17-s + (−2.85 − 0.475i)18-s + (−0.879 + 0.475i)19-s + (−0.740 − 0.804i)22-s + (−0.484 − 1.91i)24-s + (−0.677 − 0.735i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.799 + 0.600i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.799 + 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.737795944\)
\(L(\frac12)\) \(\approx\) \(1.737795944\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.879 + 0.475i)T \)
19 \( 1 + (0.879 - 0.475i)T \)
good3 \( 1 + (-1.94 + 0.324i)T + (0.945 - 0.324i)T^{2} \)
5 \( 1 + (0.677 + 0.735i)T^{2} \)
7 \( 1 + (0.401 + 0.915i)T^{2} \)
11 \( 1 + (-1.03 - 0.355i)T + (0.789 + 0.614i)T^{2} \)
13 \( 1 + (-0.945 + 0.324i)T^{2} \)
17 \( 1 + (0.439 - 0.672i)T + (-0.401 - 0.915i)T^{2} \)
23 \( 1 + (-0.945 - 0.324i)T^{2} \)
29 \( 1 + (0.401 + 0.915i)T^{2} \)
31 \( 1 + (-0.546 + 0.837i)T^{2} \)
37 \( 1 + (-0.789 - 0.614i)T^{2} \)
41 \( 1 + (0.484 - 1.91i)T + (-0.879 - 0.475i)T^{2} \)
43 \( 1 + (1.38 + 1.08i)T + (0.245 + 0.969i)T^{2} \)
47 \( 1 + (-0.789 - 0.614i)T^{2} \)
53 \( 1 + (-0.789 - 0.614i)T^{2} \)
59 \( 1 + (0.431 - 1.70i)T + (-0.879 - 0.475i)T^{2} \)
61 \( 1 + (0.986 + 0.164i)T^{2} \)
67 \( 1 + (0.156 + 1.88i)T + (-0.986 + 0.164i)T^{2} \)
71 \( 1 + (0.986 - 0.164i)T^{2} \)
73 \( 1 + (-0.268 + 0.411i)T + (-0.401 - 0.915i)T^{2} \)
79 \( 1 + (-0.245 - 0.969i)T^{2} \)
83 \( 1 + (-0.322 - 0.735i)T + (-0.677 + 0.735i)T^{2} \)
89 \( 1 + (0.439 + 0.672i)T + (-0.401 + 0.915i)T^{2} \)
97 \( 1 + (0.165 - 1.99i)T + (-0.986 - 0.164i)T^{2} \)
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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

−8.834565129168238530317256296751, −8.216835308987250601649139723669, −7.84106874253274811117460468539, −6.74655134039708445801773062614, −6.49523877097127510579054860181, −4.36688045654093608194711483881, −3.84434536975965856830578255492, −3.03341838949457437055030209676, −2.02355686081174204476383010395, −1.53600876672976608537128615534, 1.49943659804046018041924333610, 2.31352896509441490775911283471, 3.26651396097456548927851861107, 4.15989843470055544077199209325, 5.06834062321448039489319994392, 6.41394400682003987753861152150, 7.05368332757735224153067318945, 7.70854045227935374072623395044, 8.539475451365683014243716994377, 8.859359027255242834448162681815

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