Properties

Degree $2$
Conductor $123210$
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 + 8-s + 10-s − 4·11-s + 2·13-s + 16-s + 2·17-s − 4·19-s + 20-s − 4·22-s − 8·23-s + 25-s + 2·26-s − 2·29-s − 8·31-s + 32-s + 2·34-s − 4·38-s + 40-s − 10·41-s − 12·43-s − 4·44-s − 8·46-s − 7·49-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 0.554·13-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 + 0.392·26-s − 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s − 0.648·38-s + 0.158·40-s − 1.56·41-s − 1.82·43-s − 0.603·44-s − 1.17·46-s − 49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(123210\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{123210} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 123210,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.800193151\)
\(L(\frac12)\) \(\approx\) \(1.800193151\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
37 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.37313658325352, −13.15940332333068, −12.77211086873047, −12.14554887722119, −11.65684197036175, −11.22533750508136, −10.44482061179644, −10.37694102930770, −9.830263500336646, −9.169070386595709, −8.472527573328776, −8.085801878285210, −7.676837517830347, −6.896316624297528, −6.507973256185981, −5.869909737607043, −5.522032761717361, −4.974993851250380, −4.465180521015317, −3.575700258733103, −3.446009389068645, −2.575108347788565, −1.855783841880491, −1.673780090378390, −0.3250350842589707, 0.3250350842589707, 1.673780090378390, 1.855783841880491, 2.575108347788565, 3.446009389068645, 3.575700258733103, 4.465180521015317, 4.974993851250380, 5.522032761717361, 5.869909737607043, 6.507973256185981, 6.896316624297528, 7.676837517830347, 8.085801878285210, 8.472527573328776, 9.169070386595709, 9.830263500336646, 10.37694102930770, 10.44482061179644, 11.22533750508136, 11.65684197036175, 12.14554887722119, 12.77211086873047, 13.15940332333068, 13.37313658325352

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