Properties

Degree 2
Conductor $ 2^{3} \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  + 3-s − 5-s + 5·7-s − 2·9-s − 2·11-s − 13-s − 15-s − 3·17-s − 2·19-s + 5·21-s + 4·23-s − 4·25-s − 5·27-s − 6·29-s − 4·31-s − 2·33-s − 5·35-s + 11·37-s − 39-s + 8·41-s − 43-s + 2·45-s + 9·47-s + 18·49-s − 3·51-s − 12·53-s + 2·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.88·7-s − 2/3·9-s − 0.603·11-s − 0.277·13-s − 0.258·15-s − 0.727·17-s − 0.458·19-s + 1.09·21-s + 0.834·23-s − 4/5·25-s − 0.962·27-s − 1.11·29-s − 0.718·31-s − 0.348·33-s − 0.845·35-s + 1.80·37-s − 0.160·39-s + 1.24·41-s − 0.152·43-s + 0.298·45-s + 1.31·47-s + 18/7·49-s − 0.420·51-s − 1.64·53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(104\)    =    \(2^{3} \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{104} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 104,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.183611147$
$L(\frac12)$  $\approx$  $1.183611147$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−19.67137426389892, −18.56831322193301, −17.67234111781327, −16.89670961885551, −15.40290050646560, −14.77955251710581, −14.02334792908167, −12.83655769769443, −11.34302693036400, −11.04939891173548, −9.215770621891548, −8.179576985714190, −7.551388895778900, −5.513276166773824, −4.256589753556497, −2.308545077347784, 2.308545077347784, 4.256589753556497, 5.513276166773824, 7.551388895778900, 8.179576985714190, 9.215770621891548, 11.04939891173548, 11.34302693036400, 12.83655769769443, 14.02334792908167, 14.77955251710581, 15.40290050646560, 16.89670961885551, 17.67234111781327, 18.56831322193301, 19.67137426389892

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