Properties

Degree 2
Conductor $ 2^{2} \cdot 7^{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  − 3-s − 3·5-s − 2·9-s − 3·11-s − 2·13-s + 3·15-s − 3·17-s + 19-s + 3·23-s + 4·25-s + 5·27-s − 6·29-s + 7·31-s + 3·33-s − 37-s + 2·39-s − 6·41-s − 4·43-s + 6·45-s + 9·47-s + 3·51-s + 3·53-s + 9·55-s − 57-s − 9·59-s + 61-s + 6·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 2/3·9-s − 0.904·11-s − 0.554·13-s + 0.774·15-s − 0.727·17-s + 0.229·19-s + 0.625·23-s + 4/5·25-s + 0.962·27-s − 1.11·29-s + 1.25·31-s + 0.522·33-s − 0.164·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.894·45-s + 1.31·47-s + 0.420·51-s + 0.412·53-s + 1.21·55-s − 0.132·57-s − 1.17·59-s + 0.128·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(196\)    =    \(2^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{196} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 196,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.95696077874095, −19.12878000779640, −18.33405153688110, −17.29569908447036, −16.60151335857839, −15.56134818011628, −15.13752653727372, −13.85930558416162, −12.74991309686296, −11.82377978703843, −11.25257418001519, −10.31356106646674, −8.832746709943670, −7.898530976505637, −6.945688744680526, −5.512267534065264, −4.445041315837335, −2.939231782608000, 0, 2.939231782608000, 4.445041315837335, 5.512267534065264, 6.945688744680526, 7.898530976505637, 8.832746709943670, 10.31356106646674, 11.25257418001519, 11.82377978703843, 12.74991309686296, 13.85930558416162, 15.13752653727372, 15.56134818011628, 16.60151335857839, 17.29569908447036, 18.33405153688110, 19.12878000779640, 19.95696077874095

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