Properties

Label 2-2898-1.1-c1-0-36
Degree $2$
Conductor $2898$
Sign $-1$
Analytic cond. $23.1406$
Root an. cond. $4.81047$
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  − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 2·11-s − 13-s + 14-s + 16-s + 6·17-s − 6·19-s − 20-s − 2·22-s − 23-s − 4·25-s + 26-s − 28-s + 5·29-s − 8·31-s − 32-s − 6·34-s + 35-s + 3·37-s + 6·38-s + 40-s + 9·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.603·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.37·19-s − 0.223·20-s − 0.426·22-s − 0.208·23-s − 4/5·25-s + 0.196·26-s − 0.188·28-s + 0.928·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s + 0.493·37-s + 0.973·38-s + 0.158·40-s + 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2898\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(23.1406\)
Root analytic conductor: \(4.81047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2898,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−8.182202421062221226969372052166, −7.902170262885049680261636447786, −6.94545470472771959839238122839, −6.27649313876450289167807265376, −5.48165904063050717242636743713, −4.27327951186921093207944603956, −3.54654475126023085816390171508, −2.50482061136141444833796675660, −1.34263154608521710049623609300, 0, 1.34263154608521710049623609300, 2.50482061136141444833796675660, 3.54654475126023085816390171508, 4.27327951186921093207944603956, 5.48165904063050717242636743713, 6.27649313876450289167807265376, 6.94545470472771959839238122839, 7.902170262885049680261636447786, 8.182202421062221226969372052166

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