Properties

Label 2-2760-1.1-c1-0-21
Degree $2$
Conductor $2760$
Sign $1$
Analytic cond. $22.0387$
Root an. cond. $4.69454$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s + 2·11-s + 15-s + 6·17-s + 8·19-s − 23-s + 25-s + 27-s − 8·29-s − 8·31-s + 2·33-s − 2·37-s + 6·41-s + 6·43-s + 45-s − 4·47-s − 7·49-s + 6·51-s − 6·53-s + 2·55-s + 8·57-s + 2·61-s − 2·67-s − 69-s + 14·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.258·15-s + 1.45·17-s + 1.83·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s + 0.348·33-s − 0.328·37-s + 0.937·41-s + 0.914·43-s + 0.149·45-s − 0.583·47-s − 49-s + 0.840·51-s − 0.824·53-s + 0.269·55-s + 1.05·57-s + 0.256·61-s − 0.244·67-s − 0.120·69-s + 1.66·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(22.0387\)
Root analytic conductor: \(4.69454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.810756957\)
\(L(\frac12)\) \(\approx\) \(2.810756957\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.118759237493293756115638694934, −7.73052462537736614451635947593, −7.66544344625033897958161976044, −6.56418873968097065615493271281, −5.63452783598334061771707104863, −5.07807928988700731659791180838, −3.72310748489463193162600086347, −3.30360531522552296438400266076, −2.05167018221242797682190777843, −1.10373938578605422358061549806, 1.10373938578605422358061549806, 2.05167018221242797682190777843, 3.30360531522552296438400266076, 3.72310748489463193162600086347, 5.07807928988700731659791180838, 5.63452783598334061771707104863, 6.56418873968097065615493271281, 7.66544344625033897958161976044, 7.73052462537736614451635947593, 9.118759237493293756115638694934

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