Properties

Label 2-2898-1.1-c1-0-51
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 − 5·13-s + 14-s + 16-s + 2·17-s − 2·19-s − 20-s − 2·22-s − 23-s − 4·25-s − 5·26-s + 28-s − 5·29-s + 32-s + 2·34-s − 35-s − 7·37-s − 2·38-s − 40-s + 3·41-s − 11·43-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 − 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.458·19-s − 0.223·20-s − 0.426·22-s − 0.208·23-s − 4/5·25-s − 0.980·26-s + 0.188·28-s − 0.928·29-s + 0.176·32-s + 0.342·34-s − 0.169·35-s − 1.15·37-s − 0.324·38-s − 0.158·40-s + 0.468·41-s − 1.67·43-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 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−8.148768777655332185636085043607, −7.53514168418195074354126285032, −7.01337909707306634260049437654, −5.87065992920602787718470541543, −5.23633601500158405094792180873, −4.49648264852630663612761600863, −3.67995199114813514028448885568, −2.68699744522194453771645905744, −1.79797486542465053407767691029, 0, 1.79797486542465053407767691029, 2.68699744522194453771645905744, 3.67995199114813514028448885568, 4.49648264852630663612761600863, 5.23633601500158405094792180873, 5.87065992920602787718470541543, 7.01337909707306634260049437654, 7.53514168418195074354126285032, 8.148768777655332185636085043607

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