Properties

Degree $2$
Conductor $333270$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 2·13-s + 14-s + 16-s − 6·17-s + 4·19-s − 20-s + 25-s − 2·26-s − 28-s + 6·29-s − 4·31-s − 32-s + 6·34-s + 35-s − 2·37-s − 4·38-s + 40-s − 6·41-s − 8·43-s + 12·47-s + 49-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 + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.169·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-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: \(1\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.74703122534574, −12.17770668799838, −11.89839219262965, −11.38976804133623, −10.99824328430131, −10.43707897616078, −10.22815486340732, −9.530605427325487, −9.116651828345916, −8.665443953684484, −8.407227062414642, −7.762596175082262, −7.282809881486823, −6.825412036577490, −6.478658958413733, −5.961403432535544, −5.243494686650567, −4.882296049059812, −4.170336713726470, −3.535485602201853, −3.339370808827270, −2.366496309720177, −2.180924748267799, −1.226733696313042, −0.7183973268201218, 0, 0.7183973268201218, 1.226733696313042, 2.180924748267799, 2.366496309720177, 3.339370808827270, 3.535485602201853, 4.170336713726470, 4.882296049059812, 5.243494686650567, 5.961403432535544, 6.478658958413733, 6.825412036577490, 7.282809881486823, 7.762596175082262, 8.407227062414642, 8.665443953684484, 9.116651828345916, 9.530605427325487, 10.22815486340732, 10.43707897616078, 10.99824328430131, 11.38976804133623, 11.89839219262965, 12.17770668799838, 12.74703122534574

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