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  =  \((2,\ 104,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.6434240793\)
\(L(\frac12)\)  \(\approx\)  \(0.6434240793\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;13\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
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

−13.81741639926511180893815804249, −12.91148629931192100169279338687, −11.87970019424135056761325974975, −11.33427763736527423692501678920, −10.28455626039788331453937391311, −8.374845901588246109940636090068, −6.80322656561684872784900392938, −6.08438169252065874846073663176, −4.62506222520716784109149308317, −3.31991104814500528755495843356, 3.31991104814500528755495843356, 4.62506222520716784109149308317, 6.08438169252065874846073663176, 6.80322656561684872784900392938, 8.374845901588246109940636090068, 10.28455626039788331453937391311, 11.33427763736527423692501678920, 11.87970019424135056761325974975, 12.91148629931192100169279338687, 13.81741639926511180893815804249

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