Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·13-s − 25-s + 2·37-s + 49-s + 2·61-s − 2·73-s − 2·97-s − 2·109-s + ⋯
L(s)  = 1  − 2·13-s − 25-s + 2·37-s + 49-s + 2·61-s − 2·73-s − 2·97-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(144\)    =    \(2^{4} \cdot 3^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{144} (127, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 144,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.6120575525$
$L(\frac12)$  $\approx$  $0.6120575525$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 + T )^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 + T )^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( ( 1 + T )^{2} \)
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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

−13.28781709463704856973543817285, −12.30384595453317575371741865574, −11.46808950694469381236809699823, −10.13873896367981246205906970661, −9.407871242909780150626629457966, −7.980979757373726681394409609594, −7.04735552203070770115198856730, −5.59847579662198507421795787135, −4.33597269490031829437262423684, −2.51665821498373870222825682205, 2.51665821498373870222825682205, 4.33597269490031829437262423684, 5.59847579662198507421795787135, 7.04735552203070770115198856730, 7.980979757373726681394409609594, 9.407871242909780150626629457966, 10.13873896367981246205906970661, 11.46808950694469381236809699823, 12.30384595453317575371741865574, 13.28781709463704856973543817285

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