Properties

Label 2-38640-1.1-c1-0-17
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 + 2·11-s − 2·13-s + 15-s + 4·17-s + 2·19-s + 21-s + 23-s + 25-s − 27-s + 2·31-s − 2·33-s + 35-s + 10·37-s + 2·39-s + 2·41-s − 6·43-s − 45-s + 2·47-s + 49-s − 4·51-s + 6·53-s − 2·55-s − 2·57-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 + 0.970·17-s + 0.458·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.359·31-s − 0.348·33-s + 0.169·35-s + 1.64·37-s + 0.320·39-s + 0.312·41-s − 0.914·43-s − 0.149·45-s + 0.291·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s − 0.269·55-s − 0.264·57-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

\(L(1)\) \(\approx\) \(1.813042938\)
\(L(\frac12)\) \(\approx\) \(1.813042938\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−14.73667303623409, −14.53762337092354, −13.69176261587739, −13.25266613121983, −12.58720565511680, −12.15782101874600, −11.75278915240587, −11.25958379452749, −10.67544310716893, −9.961658069556645, −9.673662554186009, −9.095337062830571, −8.305642664517245, −7.773483629793769, −7.259123960506953, −6.626044717721851, −6.186368404597625, −5.390636445258831, −5.012410264792212, −4.170769833068245, −3.705217302619228, −2.968385801723718, −2.223209823975126, −1.142499532473714, −0.6004329739480276, 0.6004329739480276, 1.142499532473714, 2.223209823975126, 2.968385801723718, 3.705217302619228, 4.170769833068245, 5.012410264792212, 5.390636445258831, 6.186368404597625, 6.626044717721851, 7.259123960506953, 7.773483629793769, 8.305642664517245, 9.095337062830571, 9.673662554186009, 9.961658069556645, 10.67544310716893, 11.25958379452749, 11.75278915240587, 12.15782101874600, 12.58720565511680, 13.25266613121983, 13.69176261587739, 14.53762337092354, 14.73667303623409

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