Properties

Degree $2$
Conductor $219351$
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 − 6-s + 2·7-s + 3·8-s + 9-s − 11-s − 12-s + 2·13-s − 2·14-s − 16-s − 18-s + 2·19-s + 2·21-s + 22-s − 23-s + 3·24-s − 5·25-s − 2·26-s + 27-s − 2·28-s + 10·29-s − 4·31-s − 5·32-s − 33-s − 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.235·18-s + 0.458·19-s + 0.436·21-s + 0.213·22-s − 0.208·23-s + 0.612·24-s − 25-s − 0.392·26-s + 0.192·27-s − 0.377·28-s + 1.85·29-s − 0.718·31-s − 0.883·32-s − 0.174·33-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 219351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 219351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(219351\)    =    \(3 \cdot 11 \cdot 17^{2} \cdot 23\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{219351} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 219351,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.953116482\)
\(L(\frac12)\) \(\approx\) \(1.953116482\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 \)
23 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.05650763263157, −12.54588978829273, −12.07676354191739, −11.40825737930678, −11.02530855621859, −10.58950225500306, −9.895198757972905, −9.808374060899025, −9.152498657125573, −8.684218035696869, −8.182712524816441, −8.048583277762622, −7.477051446719413, −6.995905548112983, −6.283853663009786, −5.745437423822987, −5.061663039384013, −4.654615615869923, −4.238312584773774, −3.487136591662785, −3.126827164337362, −2.192502153054908, −1.734924666754575, −1.156942457508929, −0.4623844198464241, 0.4623844198464241, 1.156942457508929, 1.734924666754575, 2.192502153054908, 3.126827164337362, 3.487136591662785, 4.238312584773774, 4.654615615869923, 5.061663039384013, 5.745437423822987, 6.283853663009786, 6.995905548112983, 7.477051446719413, 8.048583277762622, 8.182712524816441, 8.684218035696869, 9.152498657125573, 9.808374060899025, 9.895198757972905, 10.58950225500306, 11.02530855621859, 11.40825737930678, 12.07676354191739, 12.54588978829273, 13.05650763263157

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