Properties

Label 2-162240-1.1-c1-0-177
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 4·7-s + 9-s − 4·11-s − 15-s + 6·17-s − 4·21-s + 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s + 4·33-s + 4·35-s − 2·37-s − 10·41-s − 4·43-s + 45-s + 8·47-s + 9·49-s − 6·51-s + 2·53-s − 4·55-s − 4·59-s − 14·61-s + 4·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.258·15-s + 1.45·17-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.676·35-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.840·51-s + 0.274·53-s − 0.539·55-s − 0.520·59-s − 1.79·61-s + 0.503·63-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: \(1\)
Selberg data: \((2,\ 162240,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.57939400761741, −12.93066728021694, −12.52073788071976, −12.04469534934629, −11.63434317852847, −11.03802262424380, −10.61700355641136, −10.38861064669170, −9.819707947317818, −9.202652195174414, −8.574791154793193, −8.193506244872693, −7.621717876725783, −7.333498570872979, −6.678288158121189, −6.006856573657259, −5.365807195238165, −5.178909081518382, −4.860230487891468, −4.089041166013696, −3.354810931688553, −2.779591458517302, −2.049266800247430, −1.472803089224481, −0.9718147750882420, 0, 0.9718147750882420, 1.472803089224481, 2.049266800247430, 2.779591458517302, 3.354810931688553, 4.089041166013696, 4.860230487891468, 5.178909081518382, 5.365807195238165, 6.006856573657259, 6.678288158121189, 7.333498570872979, 7.621717876725783, 8.193506244872693, 8.574791154793193, 9.202652195174414, 9.819707947317818, 10.38861064669170, 10.61700355641136, 11.03802262424380, 11.63434317852847, 12.04469534934629, 12.52073788071976, 12.93066728021694, 13.57939400761741

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