Properties

Label 2-38640-1.1-c1-0-37
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 7-s + 9-s − 2·11-s − 2·13-s − 15-s + 6·19-s + 21-s − 23-s + 25-s − 27-s − 8·29-s + 10·31-s + 2·33-s − 35-s − 10·37-s + 2·39-s − 10·41-s − 2·43-s + 45-s + 2·47-s + 49-s − 6·53-s − 2·55-s − 6·57-s + 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.258·15-s + 1.37·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.79·31-s + 0.348·33-s − 0.169·35-s − 1.64·37-s + 0.320·39-s − 1.56·41-s − 0.304·43-s + 0.149·45-s + 0.291·47-s + 1/7·49-s − 0.824·53-s − 0.269·55-s − 0.794·57-s + 0.520·59-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: \(1\)
Selberg data: \((2,\ 38640,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.32812549995219, −14.47472327280316, −13.95292877095733, −13.52217327583140, −13.00716079656807, −12.48401426925919, −11.85028655546934, −11.59158676958974, −10.85051847565471, −10.12867386860965, −10.01863658557772, −9.403552073573905, −8.756856468630896, −8.013784388018286, −7.544479611130254, −6.799994582138089, −6.516349196916861, −5.647094695278564, −5.163514207864542, −4.914294199498455, −3.806879422311120, −3.322791707415992, −2.490857826201901, −1.822951266727992, −0.8989424922409106, 0, 0.8989424922409106, 1.822951266727992, 2.490857826201901, 3.322791707415992, 3.806879422311120, 4.914294199498455, 5.163514207864542, 5.647094695278564, 6.516349196916861, 6.799994582138089, 7.544479611130254, 8.013784388018286, 8.756856468630896, 9.403552073573905, 10.01863658557772, 10.12867386860965, 10.85051847565471, 11.59158676958974, 11.85028655546934, 12.48401426925919, 13.00716079656807, 13.52217327583140, 13.95292877095733, 14.47472327280316, 15.32812549995219

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