Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7 \cdot 13 \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-s − 5-s − 7-s + 9-s + 2·11-s − 13-s + 15-s + 4·17-s + 8·19-s + 21-s − 3·23-s − 4·25-s − 27-s + 2·29-s − 4·31-s − 2·33-s + 35-s + 3·37-s + 39-s − 41-s + 5·43-s − 45-s − 6·47-s + 49-s − 4·51-s − 2·53-s − 2·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.258·15-s + 0.970·17-s + 1.83·19-s + 0.218·21-s − 0.625·23-s − 4/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.348·33-s + 0.169·35-s + 0.493·37-s + 0.160·39-s − 0.156·41-s + 0.762·43-s − 0.149·45-s − 0.875·47-s + 1/7·49-s − 0.560·51-s − 0.274·53-s − 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(179088\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13 \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{179088} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 179088,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.081330259$
$L(\frac12)$  $\approx$  $1.081330259$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;13,\;41\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;13,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 + T \)
41 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.96800949826932, −12.66827276858734, −12.06513115631575, −11.74806643432621, −11.51815698731054, −10.86982237023462, −10.25563500752110, −9.856138349965639, −9.448561934972813, −9.068736146177214, −8.221875486085398, −7.772029245416500, −7.412839789622940, −6.934566484488354, −6.277818814673584, −5.797673763482264, −5.419258024079393, −4.813610251865319, −4.155925050390199, −3.719991530868423, −3.150695684670894, −2.618257203047608, −1.563321917942310, −1.221188806576492, −0.3327941873407244, 0.3327941873407244, 1.221188806576492, 1.563321917942310, 2.618257203047608, 3.150695684670894, 3.719991530868423, 4.155925050390199, 4.813610251865319, 5.419258024079393, 5.797673763482264, 6.277818814673584, 6.934566484488354, 7.412839789622940, 7.772029245416500, 8.221875486085398, 9.068736146177214, 9.448561934972813, 9.856138349965639, 10.25563500752110, 10.86982237023462, 11.51815698731054, 11.74806643432621, 12.06513115631575, 12.66827276858734, 12.96800949826932

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