Properties

Degree $2$
Conductor $2898$
Sign $1$
Motivic weight $1$
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 − 2·13-s − 14-s + 16-s + 6·17-s + 4·19-s + 2·20-s − 4·22-s + 23-s − 25-s + 2·26-s + 28-s + 2·29-s − 8·31-s − 32-s − 6·34-s + 2·35-s + 6·37-s − 4·38-s − 2·40-s + 6·41-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 − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s + 0.208·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.338·35-s + 0.986·37-s − 0.648·38-s − 0.316·40-s + 0.937·41-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: $1$
Motivic weight: \(1\)
Character: $\chi_{2898} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2898,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.911394957\)
\(L(\frac12)\) \(\approx\) \(1.911394957\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−18.53004115564863, −18.08728210320072, −17.39043340702865, −16.89009238676353, −16.50934283341659, −15.63120825425063, −14.71010266721617, −14.34576488791879, −13.79461346534358, −12.78417402852377, −12.10550179103754, −11.57079150834365, −10.75136699363697, −10.01704046505008, −9.345746559874294, −9.129534828526019, −7.873319207026196, −7.487787413430481, −6.510476197117333, −5.799974098487750, −5.114126047719318, −3.906519771244526, −2.912897992843852, −1.809939478915623, −1.031601717570551, 1.031601717570551, 1.809939478915623, 2.912897992843852, 3.906519771244526, 5.114126047719318, 5.799974098487750, 6.510476197117333, 7.487787413430481, 7.873319207026196, 9.129534828526019, 9.345746559874294, 10.01704046505008, 10.75136699363697, 11.57079150834365, 12.10550179103754, 12.78417402852377, 13.79461346534358, 14.34576488791879, 14.71010266721617, 15.63120825425063, 16.50934283341659, 16.89009238676353, 17.39043340702865, 18.08728210320072, 18.53004115564863

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