Properties

Label 2-2898-1.1-c1-0-33
Degree $2$
Conductor $2898$
Sign $1$
Analytic cond. $23.1406$
Root an. cond. $4.81047$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2.70·5-s + 7-s + 8-s + 2.70·10-s + 4·11-s − 0.701·13-s + 14-s + 16-s − 4·17-s + 7.40·19-s + 2.70·20-s + 4·22-s − 23-s + 2.29·25-s − 0.701·26-s + 28-s − 6.70·29-s − 2·31-s + 32-s − 4·34-s + 2.70·35-s + 10.7·37-s + 7.40·38-s + 2.70·40-s + 6.70·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.20·5-s + 0.377·7-s + 0.353·8-s + 0.854·10-s + 1.20·11-s − 0.194·13-s + 0.267·14-s + 0.250·16-s − 0.970·17-s + 1.69·19-s + 0.604·20-s + 0.852·22-s − 0.208·23-s + 0.459·25-s − 0.137·26-s + 0.188·28-s − 1.24·29-s − 0.359·31-s + 0.176·32-s − 0.685·34-s + 0.456·35-s + 1.75·37-s + 1.20·38-s + 0.427·40-s + 1.04·41-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: no
Arithmetic: yes
Character: $\chi_{2898} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2898,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.181587461\)
\(L(\frac12)\) \(\approx\) \(4.181587461\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70T + 5T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 0.701T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 7.40T + 19T^{2} \)
29 \( 1 + 6.70T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 6.70T + 41T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 + 8.10T + 47T^{2} \)
53 \( 1 + 3.40T + 53T^{2} \)
59 \( 1 + 5.40T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 5.40T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 0.596T + 83T^{2} \)
89 \( 1 + 1.40T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−9.126757857835755877839062312135, −7.77559038018779645695992737940, −7.17492355305916466483826233031, −6.14073453144146507667082244266, −5.86800538294389149040174988243, −4.87896623063302869473157119088, −4.15045559389074793520254337776, −3.10296351108977347968222332467, −2.09490712769327407997606593046, −1.29393395565374479541145938594, 1.29393395565374479541145938594, 2.09490712769327407997606593046, 3.10296351108977347968222332467, 4.15045559389074793520254337776, 4.87896623063302869473157119088, 5.86800538294389149040174988243, 6.14073453144146507667082244266, 7.17492355305916466483826233031, 7.77559038018779645695992737940, 9.126757857835755877839062312135

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