Properties

Label 2-2898-1.1-c1-0-22
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 $0$

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 + 5·13-s − 14-s + 16-s + 8·19-s + 3·20-s + 23-s + 4·25-s − 5·26-s + 28-s − 3·29-s + 2·31-s − 32-s + 3·35-s − 7·37-s − 8·38-s − 3·40-s − 9·41-s − 43-s − 46-s + 3·47-s + 49-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 + 1.38·13-s − 0.267·14-s + 1/4·16-s + 1.83·19-s + 0.670·20-s + 0.208·23-s + 4/5·25-s − 0.980·26-s + 0.188·28-s − 0.557·29-s + 0.359·31-s − 0.176·32-s + 0.507·35-s − 1.15·37-s − 1.29·38-s − 0.474·40-s − 1.40·41-s − 0.152·43-s − 0.147·46-s + 0.437·47-s + 1/7·49-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: \(0\)
Selberg data: \((2,\ 2898,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.084924735\)
\(L(\frac12)\) \(\approx\) \(2.084924735\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−8.821317656689202549443528913758, −8.232524183544560860688464311829, −7.21168097014111131891212362730, −6.60811491963455317329179529961, −5.56133283177033100824366654679, −5.38147887888854304412534738693, −3.85236343837231962091875263875, −2.89077933654956654393036807338, −1.78994296405010944893448177210, −1.09857844246470733126369777529, 1.09857844246470733126369777529, 1.78994296405010944893448177210, 2.89077933654956654393036807338, 3.85236343837231962091875263875, 5.38147887888854304412534738693, 5.56133283177033100824366654679, 6.60811491963455317329179529961, 7.21168097014111131891212362730, 8.232524183544560860688464311829, 8.821317656689202549443528913758

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