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  + 2.21·3-s + 3.59·5-s − 5.06·7-s + 1.90·9-s − 2.55·11-s − 4.93·13-s + 7.96·15-s + 2.68·17-s + 4.72·19-s − 11.2·21-s − 1.49·23-s + 7.93·25-s − 2.41·27-s + 2.43·29-s + 3.19·31-s − 5.67·33-s − 18.2·35-s + 2.90·37-s − 10.9·39-s − 41-s − 7.62·43-s + 6.86·45-s + 5.15·47-s + 18.6·49-s + 5.95·51-s − 11.1·53-s − 9.20·55-s + ⋯
L(s)  = 1  + 1.27·3-s + 1.60·5-s − 1.91·7-s + 0.636·9-s − 0.771·11-s − 1.36·13-s + 2.05·15-s + 0.651·17-s + 1.08·19-s − 2.44·21-s − 0.311·23-s + 1.58·25-s − 0.465·27-s + 0.451·29-s + 0.573·31-s − 0.987·33-s − 3.07·35-s + 0.478·37-s − 1.75·39-s − 0.156·41-s − 1.16·43-s + 1.02·45-s + 0.751·47-s + 2.66·49-s + 0.833·51-s − 1.52·53-s − 1.24·55-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  =  $(2,\ 164,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.64301$
$L(\frac12)$  $\approx$  $1.64301$
$L(\frac{3}{2})$   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 \( 1 - 2.21T + 3T^{2} \)
5 \( 1 - 3.59T + 5T^{2} \)
7 \( 1 + 5.06T + 7T^{2} \)
11 \( 1 + 2.55T + 11T^{2} \)
13 \( 1 + 4.93T + 13T^{2} \)
17 \( 1 - 2.68T + 17T^{2} \)
19 \( 1 - 4.72T + 19T^{2} \)
23 \( 1 + 1.49T + 23T^{2} \)
29 \( 1 - 2.43T + 29T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 - 2.90T + 37T^{2} \)
43 \( 1 + 7.62T + 43T^{2} \)
47 \( 1 - 5.15T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 - 1.49T + 59T^{2} \)
61 \( 1 - 1.06T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 - 8.28T + 73T^{2} \)
79 \( 1 + 4.28T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 - 9.36T + 89T^{2} \)
97 \( 1 + 3.37T + 97T^{2} \)
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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

−13.15427140730918692295360380935, −12.29667576377047686997729092509, −10.03193261044176954620528220660, −9.883325739263117266406821137351, −9.113520033222656878036949681569, −7.67071754643379739839855280267, −6.48411868103710289674271580053, −5.35355840171940356639548895826, −3.17236024489173932430653019754, −2.46650905065687307909939729608, 2.46650905065687307909939729608, 3.17236024489173932430653019754, 5.35355840171940356639548895826, 6.48411868103710289674271580053, 7.67071754643379739839855280267, 9.113520033222656878036949681569, 9.883325739263117266406821137351, 10.03193261044176954620528220660, 12.29667576377047686997729092509, 13.15427140730918692295360380935

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