Properties

Label 2-2898-1.1-c1-0-29
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 + 3.70·5-s − 7-s + 8-s + 3.70·10-s + 5.70·13-s − 14-s + 16-s − 5.40·19-s + 3.70·20-s + 23-s + 8.70·25-s + 5.70·26-s − 28-s − 0.298·29-s + 6·31-s + 32-s − 3.70·35-s + 3.70·37-s − 5.40·38-s + 3.70·40-s − 7.70·41-s + 5.70·43-s + 46-s − 3.70·47-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.65·5-s − 0.377·7-s + 0.353·8-s + 1.17·10-s + 1.58·13-s − 0.267·14-s + 0.250·16-s − 1.23·19-s + 0.827·20-s + 0.208·23-s + 1.74·25-s + 1.11·26-s − 0.188·28-s − 0.0554·29-s + 1.07·31-s + 0.176·32-s − 0.625·35-s + 0.608·37-s − 0.876·38-s + 0.585·40-s − 1.20·41-s + 0.869·43-s + 0.147·46-s − 0.539·47-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

\(L(1)\) \(\approx\) \(4.141550594\)
\(L(\frac12)\) \(\approx\) \(4.141550594\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.70T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 5.70T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.40T + 19T^{2} \)
29 \( 1 + 0.298T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 3.70T + 37T^{2} \)
41 \( 1 + 7.70T + 41T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 + 3.70T + 47T^{2} \)
53 \( 1 + 9.40T + 53T^{2} \)
59 \( 1 - 0.596T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 7.40T + 71T^{2} \)
73 \( 1 + 5.40T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 2.59T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 1.10T + 97T^{2} \)
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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

−8.808923785932214279122683475355, −8.118161817542072837369374345554, −6.77536911204451311084019373597, −6.31825768553830574372918017727, −5.85857796891909436770674701139, −4.99071684813130730745196131379, −4.04923674355763669857687043844, −3.06243917093988772798819173715, −2.17680477337606366026284560933, −1.26143318420387332138780214808, 1.26143318420387332138780214808, 2.17680477337606366026284560933, 3.06243917093988772798819173715, 4.04923674355763669857687043844, 4.99071684813130730745196131379, 5.85857796891909436770674701139, 6.31825768553830574372918017727, 6.77536911204451311084019373597, 8.118161817542072837369374345554, 8.808923785932214279122683475355

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