Properties

Label 2-38640-1.1-c1-0-48
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 − 6·13-s − 15-s + 6·17-s + 4·19-s − 21-s + 23-s + 25-s + 27-s − 2·29-s − 4·31-s + 35-s + 2·37-s − 6·39-s + 2·41-s − 45-s + 4·47-s + 49-s + 6·51-s − 14·53-s + 4·57-s − 12·59-s + 2·61-s − 63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-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 − 0.718·31-s + 0.169·35-s + 0.328·37-s − 0.960·39-s + 0.312·41-s − 0.149·45-s + 0.583·47-s + 1/7·49-s + 0.840·51-s − 1.92·53-s + 0.529·57-s − 1.56·59-s + 0.256·61-s − 0.125·63-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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.98813797515399, −14.63990922480970, −14.02365227198393, −13.75963670153520, −12.80890808265788, −12.46363379934413, −12.17644185396590, −11.47238566462729, −10.88044523749930, −10.17988410199986, −9.690053243782352, −9.377098699572326, −8.784635269204509, −7.797696836108485, −7.664899006713960, −7.276175582502387, −6.454450013035041, −5.748384631645780, −5.061268688616254, −4.636028399021486, −3.729222739976865, −3.224829574039475, −2.728087923059034, −1.882285378426271, −0.9907999218270499, 0, 0.9907999218270499, 1.882285378426271, 2.728087923059034, 3.224829574039475, 3.729222739976865, 4.636028399021486, 5.061268688616254, 5.748384631645780, 6.454450013035041, 7.276175582502387, 7.664899006713960, 7.797696836108485, 8.784635269204509, 9.377098699572326, 9.690053243782352, 10.17988410199986, 10.88044523749930, 11.47238566462729, 12.17644185396590, 12.46363379934413, 12.80890808265788, 13.75963670153520, 14.02365227198393, 14.63990922480970, 14.98813797515399

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