Properties

Degree $2$
Conductor $333270$
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 − 2·13-s + 14-s + 16-s + 2·17-s + 4·19-s + 20-s − 4·22-s + 25-s + 2·26-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 + ⋯
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 − 0.554·13-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 + 1/5·25-s + 0.392·26-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333270\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{333270} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.615464923\)
\(L(\frac12)\) \(\approx\) \(1.615464923\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
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

−12.39416857152489, −12.01043852317147, −11.85506643684702, −11.03014409309940, −10.81118997022787, −10.25057344698846, −9.697932503104964, −9.359947161268837, −9.065150316083007, −8.743734398582670, −7.818357774629555, −7.513270283713805, −7.155014997172134, −6.649748978802640, −6.049020631042815, −5.543098951754239, −5.396998198790699, −4.337604392703594, −4.011949121206131, −3.377078918981015, −2.814548683201341, −2.283876298778900, −1.531542350638537, −1.205374425474863, −0.3879069295047829, 0.3879069295047829, 1.205374425474863, 1.531542350638537, 2.283876298778900, 2.814548683201341, 3.377078918981015, 4.011949121206131, 4.337604392703594, 5.396998198790699, 5.543098951754239, 6.049020631042815, 6.649748978802640, 7.155014997172134, 7.513270283713805, 7.818357774629555, 8.743734398582670, 9.065150316083007, 9.359947161268837, 9.697932503104964, 10.25057344698846, 10.81118997022787, 11.03014409309940, 11.85506643684702, 12.01043852317147, 12.39416857152489

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