Properties

Degree $2$
Conductor $346560$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 4·7-s + 9-s − 6·11-s − 4·13-s + 15-s + 6·17-s − 4·21-s + 6·23-s + 25-s + 27-s + 2·29-s − 6·33-s − 4·35-s − 8·37-s − 4·39-s − 10·41-s + 4·43-s + 45-s − 2·47-s + 9·49-s + 6·51-s + 10·53-s − 6·55-s − 12·59-s − 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.258·15-s + 1.45·17-s − 0.872·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.04·33-s − 0.676·35-s − 1.31·37-s − 0.640·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s − 0.291·47-s + 9/7·49-s + 0.840·51-s + 1.37·53-s − 0.809·55-s − 1.56·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(346560\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 19^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{346560} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 346560,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
19 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−12.75955231169202, −12.39067738185099, −12.20352554014264, −11.48846969989945, −10.57725258705274, −10.39995291339874, −10.12030442871793, −9.667814125701427, −9.200424495170186, −8.764859879084470, −8.187646975552899, −7.652336015089873, −7.238453707431341, −6.941732742962850, −6.293132704039721, −5.683140615699770, −5.280770361713578, −4.929925644042186, −4.253354421718690, −3.317826652564378, −3.139126711536175, −2.812419838267025, −2.204485846111558, −1.516626331257959, −0.6310122756055743, 0, 0.6310122756055743, 1.516626331257959, 2.204485846111558, 2.812419838267025, 3.139126711536175, 3.317826652564378, 4.253354421718690, 4.929925644042186, 5.280770361713578, 5.683140615699770, 6.293132704039721, 6.941732742962850, 7.238453707431341, 7.652336015089873, 8.187646975552899, 8.764859879084470, 9.200424495170186, 9.667814125701427, 10.12030442871793, 10.39995291339874, 10.57725258705274, 11.48846969989945, 12.20352554014264, 12.39067738185099, 12.75955231169202

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