Properties

Label 2-38640-1.1-c1-0-42
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 + 4·11-s − 2·13-s − 15-s − 6·17-s + 21-s − 23-s + 25-s − 27-s − 2·29-s − 8·31-s − 4·33-s − 35-s + 2·37-s + 2·39-s + 2·41-s + 4·43-s + 45-s + 8·47-s + 49-s + 6·51-s + 6·53-s + 4·55-s − 8·59-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 − 1.45·17-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-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 + 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 0.539·55-s − 1.04·59-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: Trivial
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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−14.97945025734039, −14.68593512516556, −13.95612403041576, −13.53380775811027, −12.96576823873608, −12.44078821491605, −11.99341552302259, −11.43414023061536, −10.77367791463153, −10.57824386554079, −9.652594449220384, −9.177822011082514, −9.069085585674030, −8.120085426094459, −7.287819668269074, −6.969209386438080, −6.328449493789736, −5.888641948134181, −5.284253214702878, −4.462720465127969, −4.090165738091516, −3.333846753458655, −2.359374358104211, −1.872737870804002, −0.9273596356821832, 0, 0.9273596356821832, 1.872737870804002, 2.359374358104211, 3.333846753458655, 4.090165738091516, 4.462720465127969, 5.284253214702878, 5.888641948134181, 6.328449493789736, 6.969209386438080, 7.287819668269074, 8.120085426094459, 9.069085585674030, 9.177822011082514, 9.652594449220384, 10.57824386554079, 10.77367791463153, 11.43414023061536, 11.99341552302259, 12.44078821491605, 12.96576823873608, 13.53380775811027, 13.95612403041576, 14.68593512516556, 14.97945025734039

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