Properties

Label 2-123840-1.1-c1-0-69
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 − 4·7-s + 2·13-s − 2·17-s + 4·19-s + 25-s + 2·29-s + 4·35-s − 10·37-s + 6·41-s − 43-s − 8·47-s + 9·49-s + 2·53-s − 2·61-s − 2·65-s − 4·67-s + 8·71-s + 10·73-s + 8·79-s − 16·83-s + 2·85-s − 6·89-s − 8·91-s − 4·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.676·35-s − 1.64·37-s + 0.937·41-s − 0.152·43-s − 1.16·47-s + 9/7·49-s + 0.274·53-s − 0.256·61-s − 0.248·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.900·79-s − 1.75·83-s + 0.216·85-s − 0.635·89-s − 0.838·91-s − 0.410·95-s + 0.203·97-s + 0.0995·101-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−13.78485782783729, −13.20302928278239, −12.89542168824406, −12.21215910874677, −12.07551800252064, −11.29947545987557, −10.92252933949973, −10.33650172042566, −9.887473446750913, −9.302403482377387, −9.082575605060069, −8.288302532588666, −7.994725508248100, −7.154869412594802, −6.802396907518384, −6.462506553797366, −5.718830859748134, −5.342341370856586, −4.544562631855002, −3.991729433263670, −3.341276220900489, −3.133767929862607, −2.366456391371888, −1.519990579451921, −0.7080475989301961, 0, 0.7080475989301961, 1.519990579451921, 2.366456391371888, 3.133767929862607, 3.341276220900489, 3.991729433263670, 4.544562631855002, 5.342341370856586, 5.718830859748134, 6.462506553797366, 6.802396907518384, 7.154869412594802, 7.994725508248100, 8.288302532588666, 9.082575605060069, 9.302403482377387, 9.887473446750913, 10.33650172042566, 10.92252933949973, 11.29947545987557, 12.07551800252064, 12.21215910874677, 12.89542168824406, 13.20302928278239, 13.78485782783729

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