Properties

Label 2-2888-1.1-c1-0-62
Degree $2$
Conductor $2888$
Sign $-1$
Analytic cond. $23.0607$
Root an. cond. $4.80216$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·7-s − 2·9-s + 2·11-s − 13-s − 5·17-s − 3·21-s − 23-s − 5·25-s + 5·27-s + 3·29-s − 4·31-s − 2·33-s − 2·37-s + 39-s + 8·41-s − 8·43-s − 8·47-s + 2·49-s + 5·51-s − 9·53-s − 59-s + 14·61-s − 6·63-s − 13·67-s + 69-s − 10·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.603·11-s − 0.277·13-s − 1.21·17-s − 0.654·21-s − 0.208·23-s − 25-s + 0.962·27-s + 0.557·29-s − 0.718·31-s − 0.348·33-s − 0.328·37-s + 0.160·39-s + 1.24·41-s − 1.21·43-s − 1.16·47-s + 2/7·49-s + 0.700·51-s − 1.23·53-s − 0.130·59-s + 1.79·61-s − 0.755·63-s − 1.58·67-s + 0.120·69-s − 1.18·71-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(23.0607\)
Root analytic conductor: \(4.80216\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2888} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2888,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.359566813592776605697943643675, −7.74613152589584229848027600152, −6.72598040565036698033673525843, −6.13690005259013972771512187418, −5.20164006899702178333342847264, −4.65444054474886627728449087886, −3.71573461775705014184176995236, −2.44767093282979086637766710358, −1.50567723507471670967120404283, 0, 1.50567723507471670967120404283, 2.44767093282979086637766710358, 3.71573461775705014184176995236, 4.65444054474886627728449087886, 5.20164006899702178333342847264, 6.13690005259013972771512187418, 6.72598040565036698033673525843, 7.74613152589584229848027600152, 8.359566813592776605697943643675

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