Properties

Degree 2
Conductor $ 2^{2} \cdot 31 $
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 − 3·5-s − 7-s + 9-s − 6·11-s + 2·13-s + 6·15-s + 6·17-s − 19-s + 2·21-s − 6·23-s + 4·25-s + 4·27-s + 31-s + 12·33-s + 3·35-s − 10·37-s − 4·39-s − 9·41-s + 8·43-s − 3·45-s − 6·49-s − 12·51-s + 18·55-s + 2·57-s − 3·59-s − 10·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 1.54·15-s + 1.45·17-s − 0.229·19-s + 0.436·21-s − 1.25·23-s + 4/5·25-s + 0.769·27-s + 0.179·31-s + 2.08·33-s + 0.507·35-s − 1.64·37-s − 0.640·39-s − 1.40·41-s + 1.21·43-s − 0.447·45-s − 6/7·49-s − 1.68·51-s + 2.42·55-s + 0.264·57-s − 0.390·59-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(124\)    =    \(2^{2} \cdot 31\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{124} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 124,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
31 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−19.24866808313792, −18.57740105828678, −17.72096750964230, −16.46994179022552, −16.01067892032290, −15.25780176290075, −13.77874419568664, −12.46026360788664, −11.98385590607492, −10.87563904560239, −10.20308922870446, −8.307924355052226, −7.509025238303075, −6.027770661270531, −4.975338092737403, −3.398231353093759, 0, 3.398231353093759, 4.975338092737403, 6.027770661270531, 7.509025238303075, 8.307924355052226, 10.20308922870446, 10.87563904560239, 11.98385590607492, 12.46026360788664, 13.77874419568664, 15.25780176290075, 16.01067892032290, 16.46994179022552, 17.72096750964230, 18.57740105828678, 19.24866808313792

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