Properties

Label 2-123840-1.1-c1-0-35
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 4·7-s − 4·11-s − 2·13-s + 6·17-s + 6·19-s − 6·23-s + 25-s + 2·29-s − 4·31-s − 4·35-s − 8·37-s + 8·41-s + 43-s − 6·47-s + 9·49-s − 6·53-s + 4·55-s + 10·61-s + 2·65-s − 12·67-s + 16·71-s + 16·73-s − 16·77-s − 4·79-s − 6·83-s − 6·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.676·35-s − 1.31·37-s + 1.24·41-s + 0.152·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s + 1.28·61-s + 0.248·65-s − 1.46·67-s + 1.89·71-s + 1.87·73-s − 1.82·77-s − 0.450·79-s − 0.658·83-s − 0.650·85-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: \(0\)
Selberg data: \((2,\ 123840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.348632723\)
\(L(\frac12)\) \(\approx\) \(2.348632723\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.84928192807026, −12.88874592292545, −12.48349764763358, −12.08778976531375, −11.61074693460181, −11.15603894499708, −10.71572159396318, −10.09306587891022, −9.790730415809786, −9.164735619746109, −8.323873544075063, −8.062061529518317, −7.698186229416579, −7.361318414691698, −6.691904134617341, −5.628418050113936, −5.516835678720278, −4.978849063708268, −4.512198908670473, −3.731806612282787, −3.245232946952862, −2.530521336211712, −1.898249220672363, −1.257076362405489, −0.4848786793911440, 0.4848786793911440, 1.257076362405489, 1.898249220672363, 2.530521336211712, 3.245232946952862, 3.731806612282787, 4.512198908670473, 4.978849063708268, 5.516835678720278, 5.628418050113936, 6.691904134617341, 7.361318414691698, 7.698186229416579, 8.062061529518317, 8.323873544075063, 9.164735619746109, 9.790730415809786, 10.09306587891022, 10.71572159396318, 11.15603894499708, 11.61074693460181, 12.08778976531375, 12.48349764763358, 12.88874592292545, 13.84928192807026

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