Properties

Label 2-8880-1.1-c1-0-75
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 + 7-s + 9-s − 3·11-s − 7·13-s + 15-s − 3·17-s + 19-s − 21-s + 3·23-s + 25-s − 27-s + 6·29-s + 10·31-s + 3·33-s − 35-s + 37-s + 7·39-s + 4·43-s − 45-s + 12·47-s − 6·49-s + 3·51-s + 9·53-s + 3·55-s − 57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.94·13-s + 0.258·15-s − 0.727·17-s + 0.229·19-s − 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.522·33-s − 0.169·35-s + 0.164·37-s + 1.12·39-s + 0.609·43-s − 0.149·45-s + 1.75·47-s − 6/7·49-s + 0.420·51-s + 1.23·53-s + 0.404·55-s − 0.132·57-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 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−7.32629650959485491042657496775, −6.88412775586520352856561830185, −6.01029824046557045434889170922, −5.08044705706523656851697759248, −4.79785182526695662763552852044, −4.12540446297713192967190181758, −2.76753372863843207958498957752, −2.44704180441384093595733820451, −1.00924948765815190675846550013, 0, 1.00924948765815190675846550013, 2.44704180441384093595733820451, 2.76753372863843207958498957752, 4.12540446297713192967190181758, 4.79785182526695662763552852044, 5.08044705706523656851697759248, 6.01029824046557045434889170922, 6.88412775586520352856561830185, 7.32629650959485491042657496775

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