Properties

Label 2-162240-1.1-c1-0-50
Degree $2$
Conductor $162240$
Sign $1$
Analytic cond. $1295.49$
Root an. cond. $35.9929$
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  − 3-s − 5-s + 2·7-s + 9-s − 4·11-s + 15-s + 8·17-s + 6·19-s − 2·21-s − 6·23-s + 25-s − 27-s + 4·29-s + 4·33-s − 2·35-s − 2·37-s + 2·41-s − 4·43-s − 45-s − 3·49-s − 8·51-s + 10·53-s + 4·55-s − 6·57-s − 4·59-s + 10·61-s + 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.258·15-s + 1.94·17-s + 1.37·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.696·33-s − 0.338·35-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.149·45-s − 3/7·49-s − 1.12·51-s + 1.37·53-s + 0.539·55-s − 0.794·57-s − 0.520·59-s + 1.28·61-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162240\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1295.49\)
Root analytic conductor: \(35.9929\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{162240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 162240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.071476739\)
\(L(\frac12)\) \(\approx\) \(2.071476739\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−13.13358998843586, −12.77721057553562, −12.08081321334381, −11.83466300699306, −11.58058759898180, −10.90704163838325, −10.31496627451034, −10.00400610605883, −9.785583460679273, −8.686726582738412, −8.476058053106464, −7.699278053978929, −7.533037752143619, −7.244166120737544, −6.141317202549193, −5.898747004835033, −5.262775068570350, −4.934456230505610, −4.448351378423053, −3.521033192443680, −3.312555514317425, −2.493552972849454, −1.770292585599334, −1.063725395463267, −0.5034466060296638, 0.5034466060296638, 1.063725395463267, 1.770292585599334, 2.493552972849454, 3.312555514317425, 3.521033192443680, 4.448351378423053, 4.934456230505610, 5.262775068570350, 5.898747004835033, 6.141317202549193, 7.244166120737544, 7.533037752143619, 7.699278053978929, 8.476058053106464, 8.686726582738412, 9.785583460679273, 10.00400610605883, 10.31496627451034, 10.90704163838325, 11.58058759898180, 11.83466300699306, 12.08081321334381, 12.77721057553562, 13.13358998843586

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