Properties

Label 2-2898-1.1-c1-0-46
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 − 3·5-s + 7-s + 8-s − 3·10-s − 4·11-s + 3·13-s + 14-s + 16-s − 3·20-s − 4·22-s + 23-s + 4·25-s + 3·26-s + 28-s − 29-s − 2·31-s + 32-s − 3·35-s − 5·37-s − 3·40-s − 5·41-s − 7·43-s − 4·44-s + 46-s + 3·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s − 1.20·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.670·20-s − 0.852·22-s + 0.208·23-s + 4/5·25-s + 0.588·26-s + 0.188·28-s − 0.185·29-s − 0.359·31-s + 0.176·32-s − 0.507·35-s − 0.821·37-s − 0.474·40-s − 0.780·41-s − 1.06·43-s − 0.603·44-s + 0.147·46-s + 0.437·47-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: $\chi_{2898} (1, \cdot )$
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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−8.198536847044129409757551013399, −7.64063675716116306855148385477, −6.97470721465790476527130888674, −5.97542202954890271986707632396, −5.12244198907696155622145254954, −4.47010972768959845276328983230, −3.59280734014224177330155858430, −2.95501883207598714731421959025, −1.62271894495161954472460503089, 0, 1.62271894495161954472460503089, 2.95501883207598714731421959025, 3.59280734014224177330155858430, 4.47010972768959845276328983230, 5.12244198907696155622145254954, 5.97542202954890271986707632396, 6.97470721465790476527130888674, 7.64063675716116306855148385477, 8.198536847044129409757551013399

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