Properties

Label 2-2898-1.1-c1-0-11
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 − 0.701·13-s − 14-s + 16-s + 7.40·19-s − 2.70·20-s + 23-s + 2.29·25-s − 0.701·26-s − 28-s − 6.70·29-s + 6·31-s + 32-s + 2.70·35-s − 2.70·37-s + 7.40·38-s − 2.70·40-s − 1.29·41-s − 0.701·43-s + 46-s + 2.70·47-s + 49-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 − 0.194·13-s − 0.267·14-s + 0.250·16-s + 1.69·19-s − 0.604·20-s + 0.208·23-s + 0.459·25-s − 0.137·26-s − 0.188·28-s − 1.24·29-s + 1.07·31-s + 0.176·32-s + 0.456·35-s − 0.444·37-s + 1.20·38-s − 0.427·40-s − 0.202·41-s − 0.106·43-s + 0.147·46-s + 0.394·47-s + 0.142·49-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: 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

\(L(1)\) \(\approx\) \(2.124506397\)
\(L(\frac12)\) \(\approx\) \(2.124506397\)
\(L(\frac{3}{2})\) 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 \( 1 + 11T^{2} \)
13 \( 1 + 0.701T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.40T + 19T^{2} \)
29 \( 1 + 6.70T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 2.70T + 37T^{2} \)
41 \( 1 + 1.29T + 41T^{2} \)
43 \( 1 + 0.701T + 43T^{2} \)
47 \( 1 - 2.70T + 47T^{2} \)
53 \( 1 - 3.40T + 53T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 5.40T + 71T^{2} \)
73 \( 1 - 7.40T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 - 2.59T + 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−8.629440024529208531499535649140, −7.77497374755217167412557347728, −7.29231799063202544499763173212, −6.55299686641013405236612339592, −5.50479754903119785543483755890, −4.89685361524396843432395793535, −3.80688298153146717399058336994, −3.45527438155497129447700702379, −2.35857044104519559859925289303, −0.802794875645493798539687683775, 0.802794875645493798539687683775, 2.35857044104519559859925289303, 3.45527438155497129447700702379, 3.80688298153146717399058336994, 4.89685361524396843432395793535, 5.50479754903119785543483755890, 6.55299686641013405236612339592, 7.29231799063202544499763173212, 7.77497374755217167412557347728, 8.629440024529208531499535649140

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