Properties

Label 2-123840-1.1-c1-0-14
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 − 2·7-s − 2·11-s + 2·13-s + 4·17-s + 6·19-s − 6·23-s + 25-s − 10·29-s − 8·31-s − 2·35-s − 2·37-s − 2·41-s + 43-s + 2·47-s − 3·49-s + 10·53-s − 2·55-s + 2·59-s + 12·61-s + 2·65-s + 12·67-s − 16·71-s + 16·73-s + 4·77-s − 8·79-s − 12·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.603·11-s + 0.554·13-s + 0.970·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s − 1.85·29-s − 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s + 0.152·43-s + 0.291·47-s − 3/7·49-s + 1.37·53-s − 0.269·55-s + 0.260·59-s + 1.53·61-s + 0.248·65-s + 1.46·67-s − 1.89·71-s + 1.87·73-s + 0.455·77-s − 0.900·79-s − 1.31·83-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: $\chi_{123840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 123840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.614548258\)
\(L(\frac12)\) \(\approx\) \(1.614548258\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.40630706214678, −13.11365733535403, −12.66068141406040, −12.17972108462965, −11.46984204915358, −11.28639056428319, −10.44022802597139, −10.12058542704739, −9.674232487598931, −9.252245082182790, −8.739193154080497, −7.950445317614411, −7.715121365856653, −7.006491946261169, −6.673940692210653, −5.694895845496284, −5.562638464399855, −5.324033365595927, −4.162819408167646, −3.696088211172650, −3.298564117233131, −2.578681444478959, −1.911533022433586, −1.285072102131984, −0.3886756584397880, 0.3886756584397880, 1.285072102131984, 1.911533022433586, 2.578681444478959, 3.298564117233131, 3.696088211172650, 4.162819408167646, 5.324033365595927, 5.562638464399855, 5.694895845496284, 6.673940692210653, 7.006491946261169, 7.715121365856653, 7.950445317614411, 8.739193154080497, 9.252245082182790, 9.674232487598931, 10.12058542704739, 10.44022802597139, 11.28639056428319, 11.46984204915358, 12.17972108462965, 12.66068141406040, 13.11365733535403, 13.40630706214678

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