Properties

Degree 2
Conductor $ 2^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0950·3-s + 1.17·5-s + 3.14·7-s − 2.99·9-s − 1.67·11-s + 6.63·13-s + 0.111·15-s + 5.16·17-s − 4.72·19-s + 0.299·21-s − 8.82·23-s − 3.63·25-s − 0.569·27-s − 1.80·29-s − 1.65·31-s − 0.159·33-s + 3.68·35-s − 1.99·37-s + 0.630·39-s − 41-s + 1.46·43-s − 3.50·45-s − 8.53·47-s + 2.89·49-s + 0.490·51-s − 9.35·53-s − 1.96·55-s + ⋯
L(s)  = 1  + 0.0549·3-s + 0.523·5-s + 1.18·7-s − 0.996·9-s − 0.505·11-s + 1.83·13-s + 0.0287·15-s + 1.25·17-s − 1.08·19-s + 0.0652·21-s − 1.83·23-s − 0.726·25-s − 0.109·27-s − 0.336·29-s − 0.298·31-s − 0.0277·33-s + 0.622·35-s − 0.327·37-s + 0.100·39-s − 0.156·41-s + 0.224·43-s − 0.521·45-s − 1.24·47-s + 0.413·49-s + 0.0687·51-s − 1.28·53-s − 0.264·55-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

\( d \)  =  \(2\)
\( N \)  =  \(164\)    =    \(2^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{164} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 164,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.30068$
$L(\frac12)$  $\approx$  $1.30068$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;41\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + T \)
good3 \( 1 - 0.0950T + 3T^{2} \)
5 \( 1 - 1.17T + 5T^{2} \)
7 \( 1 - 3.14T + 7T^{2} \)
11 \( 1 + 1.67T + 11T^{2} \)
13 \( 1 - 6.63T + 13T^{2} \)
17 \( 1 - 5.16T + 17T^{2} \)
19 \( 1 + 4.72T + 19T^{2} \)
23 \( 1 + 8.82T + 23T^{2} \)
29 \( 1 + 1.80T + 29T^{2} \)
31 \( 1 + 1.65T + 31T^{2} \)
37 \( 1 + 1.99T + 37T^{2} \)
43 \( 1 - 1.46T + 43T^{2} \)
47 \( 1 + 8.53T + 47T^{2} \)
53 \( 1 + 9.35T + 53T^{2} \)
59 \( 1 - 8.82T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 - 9.67T + 67T^{2} \)
71 \( 1 + 0.776T + 71T^{2} \)
73 \( 1 - 8.33T + 73T^{2} \)
79 \( 1 - 0.915T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + 6.44T + 89T^{2} \)
97 \( 1 + 8.32T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.94232557659217297150794226344, −11.64020616744223916685013803511, −10.97681307755426861717938235559, −9.904910083127545760255626609616, −8.410814602927378954435140271782, −8.075933363635434941303925882581, −6.16520671841520752862165626534, −5.41014431862081206521195436965, −3.75351568794207299543189756172, −1.90041449727506725045277223184, 1.90041449727506725045277223184, 3.75351568794207299543189756172, 5.41014431862081206521195436965, 6.16520671841520752862165626534, 8.075933363635434941303925882581, 8.410814602927378954435140271782, 9.904910083127545760255626609616, 10.97681307755426861717938235559, 11.64020616744223916685013803511, 12.94232557659217297150794226344

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