Properties

Label 2-123840-1.1-c1-0-111
Degree $2$
Conductor $123840$
Sign $-1$
Analytic cond. $988.867$
Root an. cond. $31.4462$
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  − 5-s + 2·7-s + 2·11-s + 2·13-s − 6·19-s + 6·23-s + 25-s − 6·29-s − 8·31-s − 2·35-s + 6·37-s + 6·41-s − 43-s − 2·47-s − 3·49-s − 14·53-s − 2·55-s + 6·59-s + 4·61-s − 2·65-s + 4·67-s + 8·71-s + 8·73-s + 4·77-s − 8·79-s − 8·83-s − 2·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s + 0.603·11-s + 0.554·13-s − 1.37·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s + 0.937·41-s − 0.152·43-s − 0.291·47-s − 3/7·49-s − 1.92·53-s − 0.269·55-s + 0.781·59-s + 0.512·61-s − 0.248·65-s + 0.488·67-s + 0.949·71-s + 0.936·73-s + 0.455·77-s − 0.900·79-s − 0.878·83-s − 0.211·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(123840\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $-1$
Analytic conductor: \(988.867\)
Root analytic conductor: \(31.4462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 123840,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
43 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.85204576718527, −13.06587487509479, −12.79606156729657, −12.56780857379529, −11.54657418910036, −11.38559808670748, −10.95855260541389, −10.66932485294134, −9.798291296539552, −9.214894857795126, −9.004520076535118, −8.230602297096054, −8.017717906110110, −7.376286232126800, −6.780148033738200, −6.416331510844233, −5.681459663745799, −5.217376361225449, −4.544012966279755, −4.093544463582015, −3.612718589442103, −2.940826452170771, −2.119233135229629, −1.606636270324658, −0.9043380751523418, 0, 0.9043380751523418, 1.606636270324658, 2.119233135229629, 2.940826452170771, 3.612718589442103, 4.093544463582015, 4.544012966279755, 5.217376361225449, 5.681459663745799, 6.416331510844233, 6.780148033738200, 7.376286232126800, 8.017717906110110, 8.230602297096054, 9.004520076535118, 9.214894857795126, 9.798291296539552, 10.66932485294134, 10.95855260541389, 11.38559808670748, 11.54657418910036, 12.56780857379529, 12.79606156729657, 13.06587487509479, 13.85204576718527

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