Properties

Label 2-12480-1.1-c1-0-14
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 − 4·7-s + 9-s + 4·11-s − 13-s + 15-s + 6·17-s + 4·21-s + 4·23-s + 25-s − 27-s + 6·29-s + 8·31-s − 4·33-s + 4·35-s + 2·37-s + 39-s + 10·41-s − 4·43-s − 45-s − 8·47-s + 9·49-s − 6·51-s + 2·53-s − 4·55-s + 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s + 1.45·17-s + 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.676·35-s + 0.328·37-s + 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.840·51-s + 0.274·53-s − 0.539·55-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

\(L(1)\) \(\approx\) \(1.413869633\)
\(L(\frac12)\) \(\approx\) \(1.413869633\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−16.26726575159374, −16.02069059962189, −15.20625945127886, −14.72021620563162, −14.06276021443453, −13.40662631091681, −12.77135093183094, −12.19729580084370, −11.94899691698993, −11.28822569958012, −10.42514679624063, −9.994253964361173, −9.443470278459418, −8.894849178152757, −8.004962797876854, −7.358567276432738, −6.605463044733187, −6.339727291221506, −5.624530264137823, −4.700869537124198, −4.105026771035838, −3.224013769789623, −2.861762228471790, −1.325314823724982, −0.6213120713653383, 0.6213120713653383, 1.325314823724982, 2.861762228471790, 3.224013769789623, 4.105026771035838, 4.700869537124198, 5.624530264137823, 6.339727291221506, 6.605463044733187, 7.358567276432738, 8.004962797876854, 8.894849178152757, 9.443470278459418, 9.994253964361173, 10.42514679624063, 11.28822569958012, 11.94899691698993, 12.19729580084370, 12.77135093183094, 13.40662631091681, 14.06276021443453, 14.72021620563162, 15.20625945127886, 16.02069059962189, 16.26726575159374

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