Properties

Label 2-2898-1.1-c1-0-16
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 − 2·5-s + 7-s + 8-s − 2·10-s + 4·11-s − 4·13-s + 14-s + 16-s + 4·17-s − 2·19-s − 2·20-s + 4·22-s − 23-s − 25-s − 4·26-s + 28-s + 6·29-s + 6·31-s + 32-s + 4·34-s − 2·35-s − 2·37-s − 2·38-s − 2·40-s + 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 1.20·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.458·19-s − 0.447·20-s + 0.852·22-s − 0.208·23-s − 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.11·29-s + 1.07·31-s + 0.176·32-s + 0.685·34-s − 0.338·35-s − 0.328·37-s − 0.324·38-s − 0.316·40-s + 1.56·41-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2898,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.711210765\)
\(L(\frac12)\) \(\approx\) \(2.711210765\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.571321321716148417122909683208, −7.88795325056025987657891193751, −7.24490800139193929221096821667, −6.48311905122979951029006016159, −5.64384097293196428736968900766, −4.62457588721377785584217918649, −4.17919756083774189926828941515, −3.28442513402038135337548479262, −2.27013855713297577192480035950, −0.940272139574865042463366049549, 0.940272139574865042463366049549, 2.27013855713297577192480035950, 3.28442513402038135337548479262, 4.17919756083774189926828941515, 4.62457588721377785584217918649, 5.64384097293196428736968900766, 6.48311905122979951029006016159, 7.24490800139193929221096821667, 7.88795325056025987657891193751, 8.571321321716148417122909683208

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