Properties

Label 2-8880-1.1-c1-0-126
Degree $2$
Conductor $8880$
Sign $-1$
Analytic cond. $70.9071$
Root an. cond. $8.42063$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 3·7-s + 9-s + 5·11-s − 2·13-s + 15-s − 7·17-s + 2·19-s − 3·21-s − 4·23-s + 25-s + 27-s − 5·29-s + 7·31-s + 5·33-s − 3·35-s + 37-s − 2·39-s + 41-s − 9·43-s + 45-s + 2·49-s − 7·51-s + 3·53-s + 5·55-s + 2·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s + 1.50·11-s − 0.554·13-s + 0.258·15-s − 1.69·17-s + 0.458·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.928·29-s + 1.25·31-s + 0.870·33-s − 0.507·35-s + 0.164·37-s − 0.320·39-s + 0.156·41-s − 1.37·43-s + 0.149·45-s + 2/7·49-s − 0.980·51-s + 0.412·53-s + 0.674·55-s + 0.264·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8880\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 37\)
Sign: $-1$
Analytic conductor: \(70.9071\)
Root analytic conductor: \(8.42063\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8880} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8880,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
37 \( 1 - T \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−7.23828983742684908835757689725, −6.60879114370641475424029472508, −6.35615556016763307942182084902, −5.38986381378098506838074134795, −4.35214417293954777865155987755, −3.89741235780868632462166144755, −2.99968757149444371759257790808, −2.28335269930114968657766798170, −1.39204236020508946559369672938, 0, 1.39204236020508946559369672938, 2.28335269930114968657766798170, 2.99968757149444371759257790808, 3.89741235780868632462166144755, 4.35214417293954777865155987755, 5.38986381378098506838074134795, 6.35615556016763307942182084902, 6.60879114370641475424029472508, 7.23828983742684908835757689725

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