Properties

Label 2-160-1.1-c3-0-8
Degree $2$
Conductor $160$
Sign $-1$
Analytic cond. $9.44030$
Root an. cond. $3.07250$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.47·3-s − 5·5-s + 31.3·7-s − 6.99·9-s + 8.94·11-s − 62·13-s + 22.3·15-s − 46·17-s − 107.·19-s − 140·21-s − 192.·23-s + 25·25-s + 152.·27-s − 90·29-s + 152.·31-s − 40.0·33-s − 156.·35-s − 214·37-s + 277.·39-s − 10·41-s + 67.0·43-s + 34.9·45-s − 398.·47-s + 637.·49-s + 205.·51-s − 678·53-s − 44.7·55-s + ⋯
L(s)  = 1  − 0.860·3-s − 0.447·5-s + 1.69·7-s − 0.259·9-s + 0.245·11-s − 1.32·13-s + 0.384·15-s − 0.656·17-s − 1.29·19-s − 1.45·21-s − 1.74·23-s + 0.200·25-s + 1.08·27-s − 0.576·29-s + 0.880·31-s − 0.211·33-s − 0.755·35-s − 0.950·37-s + 1.13·39-s − 0.0380·41-s + 0.237·43-s + 0.115·45-s − 1.23·47-s + 1.85·49-s + 0.564·51-s − 1.75·53-s − 0.109·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-1$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 5T \)
good3 \( 1 + 4.47T + 27T^{2} \)
7 \( 1 - 31.3T + 343T^{2} \)
11 \( 1 - 8.94T + 1.33e3T^{2} \)
13 \( 1 + 62T + 2.19e3T^{2} \)
17 \( 1 + 46T + 4.91e3T^{2} \)
19 \( 1 + 107.T + 6.85e3T^{2} \)
23 \( 1 + 192.T + 1.21e4T^{2} \)
29 \( 1 + 90T + 2.43e4T^{2} \)
31 \( 1 - 152.T + 2.97e4T^{2} \)
37 \( 1 + 214T + 5.06e4T^{2} \)
41 \( 1 + 10T + 6.89e4T^{2} \)
43 \( 1 - 67.0T + 7.95e4T^{2} \)
47 \( 1 + 398.T + 1.03e5T^{2} \)
53 \( 1 + 678T + 1.48e5T^{2} \)
59 \( 1 - 411.T + 2.05e5T^{2} \)
61 \( 1 - 250T + 2.26e5T^{2} \)
67 \( 1 + 49.1T + 3.00e5T^{2} \)
71 \( 1 - 366.T + 3.57e5T^{2} \)
73 \( 1 - 522T + 3.89e5T^{2} \)
79 \( 1 + 876.T + 4.93e5T^{2} \)
83 \( 1 + 380.T + 5.71e5T^{2} \)
89 \( 1 - 970T + 7.04e5T^{2} \)
97 \( 1 + 934T + 9.12e5T^{2} \)
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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

−11.72635719987079053075994375186, −11.19990040512809189244998901798, −10.18354381072050069418901439625, −8.590225515539206919452026639226, −7.80295355920318112079605761779, −6.47073839427840834831047605747, −5.14506212575153012544161861903, −4.34505219403782920223278302616, −2.05225850346205488929694359608, 0, 2.05225850346205488929694359608, 4.34505219403782920223278302616, 5.14506212575153012544161861903, 6.47073839427840834831047605747, 7.80295355920318112079605761779, 8.590225515539206919452026639226, 10.18354381072050069418901439625, 11.19990040512809189244998901798, 11.72635719987079053075994375186

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