Properties

Degree 2
Conductor $ 2^{2} \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·7-s − 3·9-s − 2·11-s − 13-s + 6·17-s − 6·19-s + 8·23-s − 25-s + 2·29-s + 10·31-s − 4·35-s − 6·37-s − 6·41-s + 4·43-s − 6·45-s − 2·47-s − 3·49-s + 6·53-s − 4·55-s − 10·59-s − 2·61-s + 6·63-s − 2·65-s + 10·67-s + 10·71-s + 2·73-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s − 9-s − 0.603·11-s − 0.277·13-s + 1.45·17-s − 1.37·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s + 1.79·31-s − 0.676·35-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.894·45-s − 0.291·47-s − 3/7·49-s + 0.824·53-s − 0.539·55-s − 1.30·59-s − 0.256·61-s + 0.755·63-s − 0.248·65-s + 1.22·67-s + 1.18·71-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(52\)    =    \(2^{2} \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{52} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 52,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8454832086$
$L(\frac12)$  $\approx$  $0.8454832086$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.51226511101540, −18.78550603197252, −17.30834521173582, −16.88570736124750, −15.42487118812446, −14.26563333906249, −13.28342306795188, −12.20226607119850, −10.63948592226509, −9.648146620493893, −8.328372177248018, −6.532556035892425, −5.331599222281810, −2.884612119728195, 2.884612119728195, 5.331599222281810, 6.532556035892425, 8.328372177248018, 9.648146620493893, 10.63948592226509, 12.20226607119850, 13.28342306795188, 14.26563333906249, 15.42487118812446, 16.88570736124750, 17.30834521173582, 18.78550603197252, 19.51226511101540

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