Properties

Degree 2
Conductor $ 2^{2} \cdot 31 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 3·7-s − 3·9-s + 6·11-s − 4·13-s − 5·19-s − 4·23-s − 4·25-s + 2·29-s − 31-s + 3·35-s − 2·37-s − 9·41-s + 2·43-s − 3·45-s + 4·47-s + 2·49-s + 12·53-s + 6·55-s + 9·59-s + 12·61-s − 9·63-s − 4·65-s − 12·67-s + 5·71-s − 14·73-s + 18·77-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s − 9-s + 1.80·11-s − 1.10·13-s − 1.14·19-s − 0.834·23-s − 4/5·25-s + 0.371·29-s − 0.179·31-s + 0.507·35-s − 0.328·37-s − 1.40·41-s + 0.304·43-s − 0.447·45-s + 0.583·47-s + 2/7·49-s + 1.64·53-s + 0.809·55-s + 1.17·59-s + 1.53·61-s − 1.13·63-s − 0.496·65-s − 1.46·67-s + 0.593·71-s − 1.63·73-s + 2.05·77-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  =  0
Selberg data  =  $(2,\ 124,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.175529561$
$L(\frac12)$  $\approx$  $1.175529561$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−19.84448497686828, −19.25314080428969, −17.71017928809132, −17.35978609376990, −16.62521143332754, −14.89137009410177, −14.51602108699238, −13.67571434204969, −12.03962739422248, −11.61492332447502, −10.30281769176056, −9.092450053143346, −8.206280855235406, −6.745871650454197, −5.524891372989489, −4.140553883914493, −2.049114250785462, 2.049114250785462, 4.140553883914493, 5.524891372989489, 6.745871650454197, 8.206280855235406, 9.092450053143346, 10.30281769176056, 11.61492332447502, 12.03962739422248, 13.67571434204969, 14.51602108699238, 14.89137009410177, 16.62521143332754, 17.35978609376990, 17.71017928809132, 19.25314080428969, 19.84448497686828

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