Properties

Label 2-38640-1.1-c1-0-34
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 + 4·19-s − 21-s − 23-s + 25-s − 27-s + 2·29-s + 8·31-s + 4·33-s − 35-s + 6·37-s − 2·39-s − 6·41-s + 4·43-s − 45-s + 49-s + 6·51-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 − 1.45·17-s + 0.917·19-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 + 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.840·51-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: \(1\)
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 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−15.34127985852297, −14.64256148441794, −13.89203365288239, −13.40897804137746, −13.17187581970933, −12.32211404726454, −11.95466628092538, −11.37249184607016, −10.88620133467872, −10.57301441989243, −9.858887982951051, −9.306238002841320, −8.531141608406281, −8.106737552861007, −7.625764606768142, −6.929189723898253, −6.402534501222229, −5.759929856529331, −5.158143769993203, −4.517032928521230, −4.206889844912944, −3.136090618165696, −2.649925622574442, −1.735511341512926, −0.8623442710072221, 0, 0.8623442710072221, 1.735511341512926, 2.649925622574442, 3.136090618165696, 4.206889844912944, 4.517032928521230, 5.158143769993203, 5.759929856529331, 6.402534501222229, 6.929189723898253, 7.625764606768142, 8.106737552861007, 8.531141608406281, 9.306238002841320, 9.858887982951051, 10.57301441989243, 10.88620133467872, 11.37249184607016, 11.95466628092538, 12.32211404726454, 13.17187581970933, 13.40897804137746, 13.89203365288239, 14.64256148441794, 15.34127985852297

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