Properties

Degree 2
Conductor $ 2^{3} \cdot 13 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 10-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 20-s − 21-s + 24-s + 26-s + 27-s + 28-s + 30-s − 2·31-s − 32-s + 34-s + 35-s + 37-s + 39-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 10-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 20-s − 21-s + 24-s + 26-s + 27-s + 28-s + 30-s − 2·31-s − 32-s + 34-s + 35-s + 37-s + 39-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(104\)    =    \(2^{3} \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{104} (51, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 104,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.3506445621$
$L(\frac12)$  $\approx$  $0.3506445621$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−14.21903403686422288216786219204, −12.69759417877772506687250298073, −11.53875891143008359430010850022, −10.91362787156601425843494571816, −9.849520919914033047471247341525, −8.768893274904782034971084298372, −7.37748796465981917417358202634, −6.13973319085271586429221786246, −5.11000336544565001207901214497, −2.09431808749519266596519118860, 2.09431808749519266596519118860, 5.11000336544565001207901214497, 6.13973319085271586429221786246, 7.37748796465981917417358202634, 8.768893274904782034971084298372, 9.849520919914033047471247341525, 10.91362787156601425843494571816, 11.53875891143008359430010850022, 12.69759417877772506687250298073, 14.21903403686422288216786219204

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