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Properties

Label 2-160-40.19-c2-0-6
Degree $2$
Conductor $160$
Sign $1$
Analytic cond. $4.35968$
Root an. cond. $2.08798$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 6·7-s + 9·9-s + 18·11-s − 6·13-s + 2·19-s + 26·23-s + 25·25-s − 30·35-s − 54·37-s − 78·41-s + 45·45-s − 86·47-s − 13·49-s + 74·53-s + 90·55-s − 78·59-s − 54·63-s − 30·65-s − 108·77-s + 81·81-s + 18·89-s + 36·91-s + 10·95-s + 162·99-s + 186·103-s + 130·115-s + ⋯
L(s)  = 1  + 5-s − 6/7·7-s + 9-s + 1.63·11-s − 0.461·13-s + 2/19·19-s + 1.13·23-s + 25-s − 6/7·35-s − 1.45·37-s − 1.90·41-s + 45-s − 1.82·47-s − 0.265·49-s + 1.39·53-s + 1.63·55-s − 1.32·59-s − 6/7·63-s − 0.461·65-s − 1.40·77-s + 81-s + 0.202·89-s + 0.395·91-s + 2/19·95-s + 1.63·99-s + 1.80·103-s + 1.13·115-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $1$
Analytic conductor: \(4.35968\)
Root analytic conductor: \(2.08798\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{160} (79, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.744768196\)
\(L(\frac12)\) \(\approx\) \(1.744768196\)
\(L(2)\) 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 \( 1 - p T \)
good3 \( ( 1 - p T )( 1 + p T ) \)
7 \( 1 + 6 T + p^{2} T^{2} \)
11 \( 1 - 18 T + p^{2} T^{2} \)
13 \( 1 + 6 T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 - 2 T + p^{2} T^{2} \)
23 \( 1 - 26 T + p^{2} T^{2} \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( ( 1 - p T )( 1 + p T ) \)
37 \( 1 + 54 T + p^{2} T^{2} \)
41 \( 1 + 78 T + p^{2} T^{2} \)
43 \( ( 1 - p T )( 1 + p T ) \)
47 \( 1 + 86 T + p^{2} T^{2} \)
53 \( 1 - 74 T + p^{2} T^{2} \)
59 \( 1 + 78 T + p^{2} T^{2} \)
61 \( ( 1 - p T )( 1 + p T ) \)
67 \( ( 1 - p T )( 1 + p T ) \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( ( 1 - p T )( 1 + p T ) \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( 1 - 18 T + p^{2} T^{2} \)
97 \( ( 1 - p T )( 1 + p T ) \)
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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

−12.76545777780297697102313863685, −11.80911274583495765497481796340, −10.36176330816047070642650555490, −9.632999908131920729047280527218, −8.882208837313737501645131560572, −6.98968377583487367604863719567, −6.46145346789182542287234731069, −4.94969732115014837939601607080, −3.42543783666814765690638074951, −1.55697769795860146057593672085, 1.55697769795860146057593672085, 3.42543783666814765690638074951, 4.94969732115014837939601607080, 6.46145346789182542287234731069, 6.98968377583487367604863719567, 8.882208837313737501645131560572, 9.632999908131920729047280527218, 10.36176330816047070642650555490, 11.80911274583495765497481796340, 12.76545777780297697102313863685

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