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  − 3.24·3-s + 2.56·5-s − 0.858·7-s + 7.49·9-s + 6.20·11-s + 1.41·13-s − 8.31·15-s − 3.93·17-s + 3.82·19-s + 2.78·21-s + 3.06·23-s + 1.58·25-s − 14.5·27-s − 8.48·29-s + 1.13·31-s − 20.1·33-s − 2.20·35-s + 8.49·37-s − 4.58·39-s − 41-s + 5.34·43-s + 19.2·45-s − 6.65·47-s − 6.26·49-s + 12.7·51-s + 6.41·53-s + 15.9·55-s + ⋯
L(s)  = 1  − 1.87·3-s + 1.14·5-s − 0.324·7-s + 2.49·9-s + 1.87·11-s + 0.392·13-s − 2.14·15-s − 0.953·17-s + 0.877·19-s + 0.607·21-s + 0.639·23-s + 0.317·25-s − 2.80·27-s − 1.57·29-s + 0.203·31-s − 3.50·33-s − 0.372·35-s + 1.39·37-s − 0.734·39-s − 0.156·41-s + 0.815·43-s + 2.86·45-s − 0.970·47-s − 0.894·49-s + 1.78·51-s + 0.881·53-s + 2.14·55-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  =  $(2,\ 164,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.833977$
$L(\frac12)$  $\approx$  $0.833977$
$L(\frac{3}{2})$   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 \( 1 + 3.24T + 3T^{2} \)
5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 + 0.858T + 7T^{2} \)
11 \( 1 - 6.20T + 11T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + 3.93T + 17T^{2} \)
19 \( 1 - 3.82T + 19T^{2} \)
23 \( 1 - 3.06T + 23T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 - 1.13T + 31T^{2} \)
37 \( 1 - 8.49T + 37T^{2} \)
43 \( 1 - 5.34T + 43T^{2} \)
47 \( 1 + 6.65T + 47T^{2} \)
53 \( 1 - 6.41T + 53T^{2} \)
59 \( 1 + 3.06T + 59T^{2} \)
61 \( 1 - 7.41T + 61T^{2} \)
67 \( 1 - 1.79T + 67T^{2} \)
71 \( 1 + 3.02T + 71T^{2} \)
73 \( 1 - 0.632T + 73T^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 + 8.76T + 83T^{2} \)
89 \( 1 + 7.89T + 89T^{2} \)
97 \( 1 - 9.86T + 97T^{2} \)
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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

−12.77004685721889666617060152103, −11.58947229657506932340571131590, −11.14357833221239082666811922028, −9.841690273657882608409876893710, −9.257187464030829018933032864878, −6.99601981099149970700711301452, −6.28520117616281367417948085015, −5.53574675057552714505672444117, −4.17490289254241671008576463925, −1.39280173733790821584173906338, 1.39280173733790821584173906338, 4.17490289254241671008576463925, 5.53574675057552714505672444117, 6.28520117616281367417948085015, 6.99601981099149970700711301452, 9.257187464030829018933032864878, 9.841690273657882608409876893710, 11.14357833221239082666811922028, 11.58947229657506932340571131590, 12.77004685721889666617060152103

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