Properties

Degree 2
Conductor $ 2^{4} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 7-s − 3·9-s + 4·11-s + 2·13-s − 6·17-s − 8·19-s − 25-s + 6·29-s − 8·31-s + 2·35-s − 2·37-s + 2·41-s + 4·43-s − 6·45-s + 8·47-s + 49-s + 6·53-s + 8·55-s − 6·61-s − 3·63-s + 4·65-s + 4·67-s + 8·71-s + 10·73-s + 4·77-s − 16·79-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 9-s + 1.20·11-s + 0.554·13-s − 1.45·17-s − 1.83·19-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.338·35-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.894·45-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 1.07·55-s − 0.768·61-s − 0.377·63-s + 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s − 1.80·79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(112\)    =    \(2^{4} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{112} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 112,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.148105728$
$L(\frac12)$  $\approx$  $1.148105728$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−19.50142871709959, −18.17903791740510, −17.33191690476361, −16.92169800647453, −15.46595920024851, −14.42477445151737, −13.79790426337419, −12.67759968692167, −11.40509490101706, −10.63587746243635, −9.117894034638209, −8.560607806232820, −6.679491946790316, −5.826132101421078, −4.184659327159711, −2.145567918024478, 2.145567918024478, 4.184659327159711, 5.826132101421078, 6.679491946790316, 8.560607806232820, 9.117894034638209, 10.63587746243635, 11.40509490101706, 12.67759968692167, 13.79790426337419, 14.42477445151737, 15.46595920024851, 16.92169800647453, 17.33191690476361, 18.17903791740510, 19.50142871709959

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