Properties

Degree $2$
Conductor $164$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.92·3-s − 3.33·5-s + 2.77·7-s + 5.58·9-s + 2.02·11-s − 3.10·13-s − 9.76·15-s − 7.91·17-s + 2.17·19-s + 8.13·21-s − 4.75·23-s + 6.10·25-s + 7.56·27-s + 3.85·29-s − 10.6·31-s + 5.94·33-s − 9.26·35-s + 6.58·37-s − 9.10·39-s − 41-s + 4.80·43-s − 18.6·45-s + 4.03·47-s + 0.719·49-s − 23.1·51-s − 1.94·53-s − 6.76·55-s + ⋯
L(s)  = 1  + 1.69·3-s − 1.49·5-s + 1.05·7-s + 1.86·9-s + 0.611·11-s − 0.862·13-s − 2.52·15-s − 1.91·17-s + 0.500·19-s + 1.77·21-s − 0.990·23-s + 1.22·25-s + 1.45·27-s + 0.716·29-s − 1.91·31-s + 1.03·33-s − 1.56·35-s + 1.08·37-s − 1.45·39-s − 0.156·41-s + 0.733·43-s − 2.77·45-s + 0.589·47-s + 0.102·49-s − 3.24·51-s − 0.266·53-s − 0.911·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{164} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59845\)
\(L(\frac12)\) \(\approx\) \(1.59845\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + T \)
good3 \( 1 - 2.92T + 3T^{2} \)
5 \( 1 + 3.33T + 5T^{2} \)
7 \( 1 - 2.77T + 7T^{2} \)
11 \( 1 - 2.02T + 11T^{2} \)
13 \( 1 + 3.10T + 13T^{2} \)
17 \( 1 + 7.91T + 17T^{2} \)
19 \( 1 - 2.17T + 19T^{2} \)
23 \( 1 + 4.75T + 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
37 \( 1 - 6.58T + 37T^{2} \)
43 \( 1 - 4.80T + 43T^{2} \)
47 \( 1 - 4.03T + 47T^{2} \)
53 \( 1 + 1.94T + 53T^{2} \)
59 \( 1 - 4.75T + 59T^{2} \)
61 \( 1 - 2.89T + 61T^{2} \)
67 \( 1 - 5.97T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + 9.24T + 73T^{2} \)
79 \( 1 + 0.320T + 79T^{2} \)
83 \( 1 + 3.69T + 83T^{2} \)
89 \( 1 - 8.96T + 89T^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−12.93363126089658526724392648885, −11.84158359511354956513819049792, −10.99084165824101185838888985919, −9.431271901955034512661160144361, −8.564538758740434850208090529504, −7.83558647351392841848529880578, −7.09038681971967530591367393449, −4.55854470282430128721053511947, −3.80460196249842388129974887510, −2.23649406422758071343303763367, 2.23649406422758071343303763367, 3.80460196249842388129974887510, 4.55854470282430128721053511947, 7.09038681971967530591367393449, 7.83558647351392841848529880578, 8.564538758740434850208090529504, 9.431271901955034512661160144361, 10.99084165824101185838888985919, 11.84158359511354956513819049792, 12.93363126089658526724392648885

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