Properties

Degree $2$
Conductor $20181$
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 − 3-s − 4-s − 2·5-s + 6-s − 7-s + 3·8-s + 9-s + 2·10-s − 4·11-s + 12-s + 2·13-s + 14-s + 2·15-s − 16-s + 6·17-s − 18-s + 4·19-s + 2·20-s + 21-s + 4·22-s − 3·24-s − 25-s − 2·26-s − 27-s + 28-s + 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.516·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.852·22-s − 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20181 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20181 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20181\)    =    \(3 \cdot 7 \cdot 31^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{20181} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 20181,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6864477934\)
\(L(\frac12)\) \(\approx\) \(0.6864477934\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
31 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−15.97093067447599, −15.39126995199303, −14.52285119261959, −14.00792551834615, −13.35715864598272, −12.92297266058436, −12.23295666223598, −11.85122066741267, −11.11612347276038, −10.53003521855545, −10.16902242585754, −9.564507211127172, −8.973757692144234, −8.138604069433964, −7.815843157674564, −7.430999783093896, −6.605596004230686, −5.722691954859582, −5.204438932435195, −4.692622599457530, −3.599218076645126, −3.499498522694489, −2.229709283700652, −1.050975180780795, −0.5048522018686031, 0.5048522018686031, 1.050975180780795, 2.229709283700652, 3.499498522694489, 3.599218076645126, 4.692622599457530, 5.204438932435195, 5.722691954859582, 6.605596004230686, 7.430999783093896, 7.815843157674564, 8.138604069433964, 8.973757692144234, 9.564507211127172, 10.16902242585754, 10.53003521855545, 11.11612347276038, 11.85122066741267, 12.23295666223598, 12.92297266058436, 13.35715864598272, 14.00792551834615, 14.52285119261959, 15.39126995199303, 15.97093067447599

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