Properties

Degree $2$
Conductor $141120$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·11-s + 2·13-s − 4·17-s − 8·23-s + 25-s + 2·31-s − 8·37-s − 2·41-s + 2·43-s + 10·47-s − 2·53-s − 2·55-s − 4·59-s − 10·61-s − 2·65-s − 2·67-s + 12·71-s − 10·73-s + 16·79-s − 16·83-s + 4·85-s + 14·89-s − 6·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.603·11-s + 0.554·13-s − 0.970·17-s − 1.66·23-s + 1/5·25-s + 0.359·31-s − 1.31·37-s − 0.312·41-s + 0.304·43-s + 1.45·47-s − 0.274·53-s − 0.269·55-s − 0.520·59-s − 1.28·61-s − 0.248·65-s − 0.244·67-s + 1.42·71-s − 1.17·73-s + 1.80·79-s − 1.75·83-s + 0.433·85-s + 1.48·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141120\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{141120} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 141120,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.71352808979361, −13.20416091763940, −12.58336164060171, −12.16192251608699, −11.76616839788541, −11.32034516996126, −10.73876708558826, −10.36533144518280, −9.838086348852033, −9.138855923758446, −8.781931454743165, −8.422695612840112, −7.660514050267677, −7.416562546955929, −6.655374910620733, −6.213380345691828, −5.888381037897045, −5.006714125994987, −4.576174917772013, −3.887256818924012, −3.673800938727251, −2.881947323751819, −2.115276041890186, −1.652512399413405, −0.7684745816822794, 0, 0.7684745816822794, 1.652512399413405, 2.115276041890186, 2.881947323751819, 3.673800938727251, 3.887256818924012, 4.576174917772013, 5.006714125994987, 5.888381037897045, 6.213380345691828, 6.655374910620733, 7.416562546955929, 7.660514050267677, 8.422695612840112, 8.781931454743165, 9.138855923758446, 9.838086348852033, 10.36533144518280, 10.73876708558826, 11.32034516996126, 11.76616839788541, 12.16192251608699, 12.58336164060171, 13.20416091763940, 13.71352808979361

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