Properties

Degree $2$
Conductor $106470$
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 + 5-s − 7-s + 8-s + 10-s + 4·11-s − 14-s + 16-s − 2·17-s + 4·19-s + 20-s + 4·22-s + 8·23-s + 25-s − 28-s − 6·29-s + 8·31-s + 32-s − 2·34-s − 35-s + 2·37-s + 4·38-s + 40-s + 2·41-s − 12·43-s + 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.169·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.312·41-s − 1.82·43-s + 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(106470\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{106470} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 106470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.599331739\)
\(L(\frac12)\) \(\approx\) \(5.599331739\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
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

−13.62815564050214, −13.32314278190866, −12.77876679002348, −12.31741619540823, −11.73438009454374, −11.29194348723269, −11.02925924367918, −10.19430720131458, −9.738758920205815, −9.329718573998970, −8.849258920230444, −8.223067532102505, −7.538903768444729, −6.980191697273924, −6.486854125020690, −6.262400910718289, −5.491332240452589, −4.897470179105863, −4.603828823046403, −3.697564683629968, −3.331565979219211, −2.786874943604142, −1.968680692984941, −1.369418673737635, −0.6726529825668873, 0.6726529825668873, 1.369418673737635, 1.968680692984941, 2.786874943604142, 3.331565979219211, 3.697564683629968, 4.603828823046403, 4.897470179105863, 5.491332240452589, 6.262400910718289, 6.486854125020690, 6.980191697273924, 7.538903768444729, 8.223067532102505, 8.849258920230444, 9.329718573998970, 9.738758920205815, 10.19430720131458, 11.02925924367918, 11.29194348723269, 11.73438009454374, 12.31741619540823, 12.77876679002348, 13.32314278190866, 13.62815564050214

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