Properties

Label 2-38640-1.1-c1-0-1
Degree $2$
Conductor $38640$
Sign $1$
Analytic cond. $308.541$
Root an. cond. $17.5653$
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 − 7-s + 9-s − 4·11-s − 2·13-s + 15-s − 2·17-s − 4·19-s + 21-s + 23-s + 25-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 35-s − 2·37-s + 2·39-s + 2·41-s − 45-s − 4·47-s + 49-s + 2·51-s + 6·53-s + 4·55-s + 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.169·35-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38640\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(308.541\)
Root analytic conductor: \(17.5653\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{38640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5441034010\)
\(L(\frac12)\) \(\approx\) \(0.5441034010\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−15.07300657213900, −14.32559433121431, −13.61999849442444, −13.23114559099882, −12.68250895326053, −12.22240396241990, −11.75612466874358, −11.12251043337547, −10.58887113547310, −10.15809743284049, −9.763651652239665, −8.811095819879676, −8.438192321731841, −7.811523592716668, −7.222658692036723, −6.642175896162881, −6.183747003300467, −5.424906733829048, −4.827735982854916, −4.417124818806420, −3.667307405024059, −2.687777224637560, −2.461267735757800, −1.245591680325796, −0.2958425280373538, 0.2958425280373538, 1.245591680325796, 2.461267735757800, 2.687777224637560, 3.667307405024059, 4.417124818806420, 4.827735982854916, 5.424906733829048, 6.183747003300467, 6.642175896162881, 7.222658692036723, 7.811523592716668, 8.438192321731841, 8.811095819879676, 9.763651652239665, 10.15809743284049, 10.58887113547310, 11.12251043337547, 11.75612466874358, 12.22240396241990, 12.68250895326053, 13.23114559099882, 13.61999849442444, 14.32559433121431, 15.07300657213900

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