Properties

Label 2-86640-1.1-c1-0-48
Degree $2$
Conductor $86640$
Sign $-1$
Analytic cond. $691.823$
Root an. cond. $26.3025$
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 + 9-s − 4·11-s − 2·13-s + 15-s + 2·17-s − 4·23-s + 25-s − 27-s − 6·29-s + 4·31-s + 4·33-s + 6·37-s + 2·39-s − 10·41-s + 4·43-s − 45-s + 12·47-s − 7·49-s − 2·51-s − 6·53-s + 4·55-s − 12·59-s − 2·61-s + 2·65-s + 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s + 0.986·37-s + 0.320·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 1.75·47-s − 49-s − 0.280·51-s − 0.824·53-s + 0.539·55-s − 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(86640\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(691.823\)
Root analytic conductor: \(26.3025\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{86640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 86640,\ (\ :1/2),\ -1)\)

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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.10495596215027, −13.61621940519941, −13.11710447022055, −12.44128127236872, −12.35837293453863, −11.64784175634347, −11.24299324867990, −10.61722447185268, −10.35493378967841, −9.617972263118206, −9.405174912962410, −8.409547508265703, −8.067551605684031, −7.516706233475963, −7.203671440983774, −6.398158017018184, −5.867151898210536, −5.419929980096095, −4.756902163769971, −4.406490404119228, −3.598784742985754, −3.024544810737023, −2.327397399890202, −1.653683914387751, −0.6654256230701555, 0, 0.6654256230701555, 1.653683914387751, 2.327397399890202, 3.024544810737023, 3.598784742985754, 4.406490404119228, 4.756902163769971, 5.419929980096095, 5.867151898210536, 6.398158017018184, 7.203671440983774, 7.516706233475963, 8.067551605684031, 8.409547508265703, 9.405174912962410, 9.617972263118206, 10.35493378967841, 10.61722447185268, 11.24299324867990, 11.64784175634347, 12.35837293453863, 12.44128127236872, 13.11710447022055, 13.61621940519941, 14.10495596215027

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