Properties

Degree 2
Conductor $ 7^{2} $
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-s − 4-s − 3·8-s − 3·9-s + 4·11-s − 16-s − 3·18-s + 4·22-s + 8·23-s − 5·25-s + 2·29-s + 5·32-s + 3·36-s − 6·37-s − 12·43-s − 4·44-s + 8·46-s − 5·50-s − 10·53-s + 2·58-s + 7·64-s + 4·67-s + 16·71-s + 9·72-s − 6·74-s + 8·79-s + 9·81-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 9-s + 1.20·11-s − 1/4·16-s − 0.707·18-s + 0.852·22-s + 1.66·23-s − 25-s + 0.371·29-s + 0.883·32-s + 1/2·36-s − 0.986·37-s − 1.82·43-s − 0.603·44-s + 1.17·46-s − 0.707·50-s − 1.37·53-s + 0.262·58-s + 7/8·64-s + 0.488·67-s + 1.89·71-s + 1.06·72-s − 0.697·74-s + 0.900·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(49\)    =    \(7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{49} (1, \cdot )$
Sato-Tate  :  $N(\mathrm{U}(1))$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 49,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9666558528$
$L(\frac12)$  $\approx$  $0.9666558528$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 7$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 7$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−19.89261077892183, −18.93330835276050, −17.58953823246936, −16.91414841722080, −15.25649719028728, −14.36789003254179, −13.55829126515042, −12.27943344760253, −11.30850265031875, −9.489525085039792, −8.498120181782134, −6.478036595894264, −5.086734638147837, −3.457739849416041, 3.457739849416041, 5.086734638147837, 6.478036595894264, 8.498120181782134, 9.489525085039792, 11.30850265031875, 12.27943344760253, 13.55829126515042, 14.36789003254179, 15.25649719028728, 16.91414841722080, 17.58953823246936, 18.93330835276050, 19.89261077892183

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