Properties

Label 2-8880-1.1-c1-0-107
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 − 4·7-s + 9-s + 4·11-s + 2·13-s − 15-s + 2·17-s + 2·19-s − 4·21-s − 6·23-s + 25-s + 27-s − 6·29-s − 10·31-s + 4·33-s + 4·35-s − 37-s + 2·39-s − 2·41-s + 2·43-s − 45-s + 4·47-s + 9·49-s + 2·51-s + 10·53-s − 4·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.458·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.79·31-s + 0.696·33-s + 0.676·35-s − 0.164·37-s + 0.320·39-s − 0.312·41-s + 0.304·43-s − 0.149·45-s + 0.583·47-s + 9/7·49-s + 0.280·51-s + 1.37·53-s − 0.539·55-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: Trivial
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.21926630378350476268335375684, −6.96508549557073085237015466049, −5.98084639911729352920572736576, −5.61026164758803449801977151336, −4.20611857983035973809175420302, −3.69238914280008987422092049833, −3.36269887825618013694118670117, −2.27366356695913427981582014377, −1.25162873873077222773873005415, 0, 1.25162873873077222773873005415, 2.27366356695913427981582014377, 3.36269887825618013694118670117, 3.69238914280008987422092049833, 4.20611857983035973809175420302, 5.61026164758803449801977151336, 5.98084639911729352920572736576, 6.96508549557073085237015466049, 7.21926630378350476268335375684

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