Properties

Label 2-55-1.1-c5-0-3
Degree $2$
Conductor $55$
Sign $1$
Analytic cond. $8.82111$
Root an. cond. $2.97003$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.73·2-s − 29.5·3-s − 24.5·4-s − 25·5-s − 80.9·6-s + 175.·7-s − 154.·8-s + 632.·9-s − 68.3·10-s + 121·11-s + 725.·12-s − 441.·13-s + 481.·14-s + 739.·15-s + 361.·16-s + 340.·17-s + 1.72e3·18-s − 960.·19-s + 612.·20-s − 5.20e3·21-s + 330.·22-s + 1.30e3·23-s + 4.57e3·24-s + 625·25-s − 1.20e3·26-s − 1.15e4·27-s − 4.31e3·28-s + ⋯
L(s)  = 1  + 0.483·2-s − 1.89·3-s − 0.766·4-s − 0.447·5-s − 0.917·6-s + 1.35·7-s − 0.854·8-s + 2.60·9-s − 0.216·10-s + 0.301·11-s + 1.45·12-s − 0.724·13-s + 0.655·14-s + 0.848·15-s + 0.353·16-s + 0.285·17-s + 1.25·18-s − 0.610·19-s + 0.342·20-s − 2.57·21-s + 0.145·22-s + 0.514·23-s + 1.62·24-s + 0.200·25-s − 0.350·26-s − 3.03·27-s − 1.03·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $1$
Analytic conductor: \(8.82111\)
Root analytic conductor: \(2.97003\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8881749871\)
\(L(\frac12)\) \(\approx\) \(0.8881749871\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 25T \)
11 \( 1 - 121T \)
good2 \( 1 - 2.73T + 32T^{2} \)
3 \( 1 + 29.5T + 243T^{2} \)
7 \( 1 - 175.T + 1.68e4T^{2} \)
13 \( 1 + 441.T + 3.71e5T^{2} \)
17 \( 1 - 340.T + 1.41e6T^{2} \)
19 \( 1 + 960.T + 2.47e6T^{2} \)
23 \( 1 - 1.30e3T + 6.43e6T^{2} \)
29 \( 1 - 7.02e3T + 2.05e7T^{2} \)
31 \( 1 - 7.34e3T + 2.86e7T^{2} \)
37 \( 1 - 9.92e3T + 6.93e7T^{2} \)
41 \( 1 - 1.04e4T + 1.15e8T^{2} \)
43 \( 1 + 7.00e3T + 1.47e8T^{2} \)
47 \( 1 + 2.68e4T + 2.29e8T^{2} \)
53 \( 1 + 1.54e4T + 4.18e8T^{2} \)
59 \( 1 - 1.66e4T + 7.14e8T^{2} \)
61 \( 1 - 1.07e3T + 8.44e8T^{2} \)
67 \( 1 + 1.18e4T + 1.35e9T^{2} \)
71 \( 1 - 7.67e4T + 1.80e9T^{2} \)
73 \( 1 - 7.31e4T + 2.07e9T^{2} \)
79 \( 1 + 7.19e4T + 3.07e9T^{2} \)
83 \( 1 - 1.91e4T + 3.93e9T^{2} \)
89 \( 1 - 4.26e4T + 5.58e9T^{2} \)
97 \( 1 + 4.88e4T + 8.58e9T^{2} \)
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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

−14.34036499246723941747289978850, −12.81534700303184943638105894047, −11.95671137919171856141557479934, −11.21224488234256247487713498141, −9.939898502645879939190774206484, −8.065135576433019066855067755115, −6.42800852916647757865537717115, −4.99966082622999785851961049667, −4.48859712253087428608424215645, −0.828451854330644962842995740742, 0.828451854330644962842995740742, 4.48859712253087428608424215645, 4.99966082622999785851961049667, 6.42800852916647757865537717115, 8.065135576433019066855067755115, 9.939898502645879939190774206484, 11.21224488234256247487713498141, 11.95671137919171856141557479934, 12.81534700303184943638105894047, 14.34036499246723941747289978850

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