Properties

Degree 2
Conductor $ 2^{4} \cdot 7 \cdot 19^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 7-s + 9-s + 4·13-s + 6·17-s + 2·21-s − 5·25-s + 4·27-s + 6·29-s − 4·31-s − 2·37-s − 8·39-s − 6·41-s − 8·43-s + 12·47-s + 49-s − 12·51-s − 6·53-s − 6·59-s + 8·61-s − 63-s − 4·67-s + 2·73-s + 10·75-s + 8·79-s − 11·81-s + 6·83-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s + 0.436·21-s − 25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.328·37-s − 1.28·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s − 1.68·51-s − 0.824·53-s − 0.781·59-s + 1.02·61-s − 0.125·63-s − 0.488·67-s + 0.234·73-s + 1.15·75-s + 0.900·79-s − 1.22·81-s + 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(40432\)    =    \(2^{4} \cdot 7 \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{40432} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 40432,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
19 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.22437912574808, −14.33694597826733, −14.05666433932757, −13.39445857838357, −12.92774111560071, −12.13278901069753, −12.01162075041438, −11.48644387831505, −10.75718875509226, −10.45516828843729, −9.939977188575515, −9.276556159674474, −8.633088081916723, −8.081070482945496, −7.474091172021703, −6.754395407596197, −6.214852625893674, −5.865076970491468, −5.237198526702626, −4.777803882138078, −3.723796896271927, −3.496729518719076, −2.566586717205066, −1.531143922978988, −0.9050614943608525, 0, 0.9050614943608525, 1.531143922978988, 2.566586717205066, 3.496729518719076, 3.723796896271927, 4.777803882138078, 5.237198526702626, 5.865076970491468, 6.214852625893674, 6.754395407596197, 7.474091172021703, 8.081070482945496, 8.633088081916723, 9.276556159674474, 9.939977188575515, 10.45516828843729, 10.75718875509226, 11.48644387831505, 12.01162075041438, 12.13278901069753, 12.92774111560071, 13.39445857838357, 14.05666433932757, 14.33694597826733, 15.22437912574808

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