Properties

Label 2-160-1.1-c5-0-7
Degree $2$
Conductor $160$
Sign $1$
Analytic cond. $25.6614$
Root an. cond. $5.06570$
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  + 13.4·3-s − 25·5-s + 138.·7-s − 62.9·9-s + 259.·11-s + 154·13-s − 335.·15-s + 178·17-s + 965.·19-s + 1.86e3·21-s + 2.63e3·23-s + 625·25-s − 4.10e3·27-s + 4.11e3·29-s + 3.15e3·31-s + 3.48e3·33-s − 3.46e3·35-s + 7.44e3·37-s + 2.06e3·39-s + 7.27e3·41-s − 1.79e4·43-s + 1.57e3·45-s − 7.41e3·47-s + 2.41e3·49-s + 2.38e3·51-s + 3.22e4·53-s − 6.48e3·55-s + ⋯
L(s)  = 1  + 0.860·3-s − 0.447·5-s + 1.06·7-s − 0.259·9-s + 0.646·11-s + 0.252·13-s − 0.384·15-s + 0.149·17-s + 0.613·19-s + 0.920·21-s + 1.03·23-s + 0.200·25-s − 1.08·27-s + 0.907·29-s + 0.590·31-s + 0.556·33-s − 0.478·35-s + 0.893·37-s + 0.217·39-s + 0.675·41-s − 1.47·43-s + 0.115·45-s − 0.489·47-s + 0.143·49-s + 0.128·51-s + 1.57·53-s − 0.289·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $1$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.831465624\)
\(L(\frac12)\) \(\approx\) \(2.831465624\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 25T \)
good3 \( 1 - 13.4T + 243T^{2} \)
7 \( 1 - 138.T + 1.68e4T^{2} \)
11 \( 1 - 259.T + 1.61e5T^{2} \)
13 \( 1 - 154T + 3.71e5T^{2} \)
17 \( 1 - 178T + 1.41e6T^{2} \)
19 \( 1 - 965.T + 2.47e6T^{2} \)
23 \( 1 - 2.63e3T + 6.43e6T^{2} \)
29 \( 1 - 4.11e3T + 2.05e7T^{2} \)
31 \( 1 - 3.15e3T + 2.86e7T^{2} \)
37 \( 1 - 7.44e3T + 6.93e7T^{2} \)
41 \( 1 - 7.27e3T + 1.15e8T^{2} \)
43 \( 1 + 1.79e4T + 1.47e8T^{2} \)
47 \( 1 + 7.41e3T + 2.29e8T^{2} \)
53 \( 1 - 3.22e4T + 4.18e8T^{2} \)
59 \( 1 - 3.40e4T + 7.14e8T^{2} \)
61 \( 1 - 2.67e4T + 8.44e8T^{2} \)
67 \( 1 - 4.98e4T + 1.35e9T^{2} \)
71 \( 1 - 5.41e4T + 1.80e9T^{2} \)
73 \( 1 + 1.85e4T + 2.07e9T^{2} \)
79 \( 1 + 8.67e4T + 3.07e9T^{2} \)
83 \( 1 + 7.86e4T + 3.93e9T^{2} \)
89 \( 1 + 1.07e5T + 5.58e9T^{2} \)
97 \( 1 + 1.08e5T + 8.58e9T^{2} \)
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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

−11.74816590552454208981111980629, −11.24064616394098973433284354837, −9.797978384422195190417687725242, −8.626722585946680871376182849515, −8.104509441801592789407893297161, −6.89792846127612672238797798937, −5.28268576079992720684403837334, −3.99615474617049886830672775601, −2.72603520867586049627154964342, −1.15238636235254227102328500260, 1.15238636235254227102328500260, 2.72603520867586049627154964342, 3.99615474617049886830672775601, 5.28268576079992720684403837334, 6.89792846127612672238797798937, 8.104509441801592789407893297161, 8.626722585946680871376182849515, 9.797978384422195190417687725242, 11.24064616394098973433284354837, 11.74816590552454208981111980629

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