Properties

Label 2-12480-1.1-c1-0-41
Degree $2$
Conductor $12480$
Sign $1$
Analytic cond. $99.6533$
Root an. cond. $9.98265$
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 + 2·7-s + 9-s − 4·11-s + 13-s + 15-s + 8·17-s + 6·19-s + 2·21-s + 6·23-s + 25-s + 27-s + 4·29-s − 4·33-s + 2·35-s + 2·37-s + 39-s − 2·41-s + 4·43-s + 45-s − 3·49-s + 8·51-s + 10·53-s − 4·55-s + 6·57-s − 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.258·15-s + 1.94·17-s + 1.37·19-s + 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.696·33-s + 0.338·35-s + 0.328·37-s + 0.160·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 3/7·49-s + 1.12·51-s + 1.37·53-s − 0.539·55-s + 0.794·57-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12480\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(99.6533\)
Root analytic conductor: \(9.98265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12480} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 12480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.917309519\)
\(L(\frac12)\) \(\approx\) \(3.917309519\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−16.28223366278880, −15.77837853363169, −15.02284783140895, −14.64304309119778, −14.03837006402856, −13.58466224577333, −13.04505358543096, −12.37348102077284, −11.79827576628973, −11.11020627842950, −10.42364750911476, −9.979007657982605, −9.421915140763862, −8.600382147573920, −8.139720506923575, −7.465809903468753, −7.148922620564237, −5.966123607983562, −5.349767738328160, −5.007035876305118, −4.026470511258898, −2.942340418721057, −2.855108192012541, −1.550225786852449, −0.9655683628813539, 0.9655683628813539, 1.550225786852449, 2.855108192012541, 2.942340418721057, 4.026470511258898, 5.007035876305118, 5.349767738328160, 5.966123607983562, 7.148922620564237, 7.465809903468753, 8.139720506923575, 8.600382147573920, 9.421915140763862, 9.979007657982605, 10.42364750911476, 11.11020627842950, 11.79827576628973, 12.37348102077284, 13.04505358543096, 13.58466224577333, 14.03837006402856, 14.64304309119778, 15.02284783140895, 15.77837853363169, 16.28223366278880

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