Properties

Degree 2
Conductor $ 2^{2} \cdot 37 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s − 3·7-s − 2·9-s + 5·11-s + 4·15-s − 6·17-s + 2·19-s + 3·21-s − 6·23-s + 11·25-s + 5·27-s − 6·29-s + 4·31-s − 5·33-s + 12·35-s + 37-s − 9·41-s + 4·43-s + 8·45-s − 7·47-s + 2·49-s + 6·51-s + 9·53-s − 20·55-s − 2·57-s − 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s − 1.13·7-s − 2/3·9-s + 1.50·11-s + 1.03·15-s − 1.45·17-s + 0.458·19-s + 0.654·21-s − 1.25·23-s + 11/5·25-s + 0.962·27-s − 1.11·29-s + 0.718·31-s − 0.870·33-s + 2.02·35-s + 0.164·37-s − 1.40·41-s + 0.609·43-s + 1.19·45-s − 1.02·47-s + 2/7·49-s + 0.840·51-s + 1.23·53-s − 2.69·55-s − 0.264·57-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(148\)    =    \(2^{2} \cdot 37\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{148} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 148,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.82392593429577, −19.28779961443637, −18.14838177506053, −16.97477842346019, −16.31879754934654, −15.57451293078973, −14.68126527562089, −13.41182482917898, −12.04878843613670, −11.80292650710755, −10.84165866699063, −9.340745919593799, −8.373681526994174, −7.028424851814057, −6.195615824773030, −4.374191941510800, −3.392475834528531, 0, 3.392475834528531, 4.374191941510800, 6.195615824773030, 7.028424851814057, 8.373681526994174, 9.340745919593799, 10.84165866699063, 11.80292650710755, 12.04878843613670, 13.41182482917898, 14.68126527562089, 15.57451293078973, 16.31879754934654, 16.97477842346019, 18.14838177506053, 19.28779961443637, 19.82392593429577

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