Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 94·7-s + 81·9-s + 146·13-s − 46·19-s − 846·21-s + 625·25-s + 729·27-s + 194·31-s − 2.06e3·37-s + 1.31e3·39-s − 3.21e3·43-s + 6.43e3·49-s − 414·57-s − 1.96e3·61-s − 7.61e3·63-s + 5.90e3·67-s − 8.54e3·73-s + 5.62e3·75-s + 7.68e3·79-s + 6.56e3·81-s − 1.37e4·91-s + 1.74e3·93-s − 1.88e4·97-s + 1.64e4·103-s + 2.20e4·109-s − 1.85e4·111-s + ⋯
L(s)  = 1  + 3-s − 1.91·7-s + 9-s + 0.863·13-s − 0.127·19-s − 1.91·21-s + 25-s + 27-s + 0.201·31-s − 1.50·37-s + 0.863·39-s − 1.73·43-s + 2.68·49-s − 0.127·57-s − 0.528·61-s − 1.91·63-s + 1.31·67-s − 1.60·73-s + 75-s + 1.23·79-s + 81-s − 1.65·91-s + 0.201·93-s − 1.99·97-s + 1.54·103-s + 1.85·109-s − 1.50·111-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  $\chi_{12} (5, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :2),\ 1)$
$L(\frac{5}{2})$  $\approx$  $1.27025$
$L(\frac12)$  $\approx$  $1.27025$
$L(3)$   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 \( 1 - p^{2} T \)
good5 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
7 \( 1 + 94 T + p^{4} T^{2} \)
11 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
13 \( 1 - 146 T + p^{4} T^{2} \)
17 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
19 \( 1 + 46 T + p^{4} T^{2} \)
23 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
29 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
31 \( 1 - 194 T + p^{4} T^{2} \)
37 \( 1 + 2062 T + p^{4} T^{2} \)
41 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
43 \( 1 + 3214 T + p^{4} T^{2} \)
47 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
53 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
59 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
61 \( 1 + 1966 T + p^{4} T^{2} \)
67 \( 1 - 5906 T + p^{4} T^{2} \)
71 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
73 \( 1 + 8542 T + p^{4} T^{2} \)
79 \( 1 - 7682 T + p^{4} T^{2} \)
83 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
89 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
97 \( 1 + 18814 T + p^{4} T^{2} \)
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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

−19.48202824504214113353894664780, −18.58739947477216633629722604007, −16.42249185595686761177404294704, −15.40289263271359717839663668376, −13.68117599380015502447117668360, −12.67522958378178130451058744024, −10.15212070800515013248741148543, −8.826023848599939488989504935568, −6.71607215381950625341138839890, −3.35298978555788219141257198948, 3.35298978555788219141257198948, 6.71607215381950625341138839890, 8.826023848599939488989504935568, 10.15212070800515013248741148543, 12.67522958378178130451058744024, 13.68117599380015502447117668360, 15.40289263271359717839663668376, 16.42249185595686761177404294704, 18.58739947477216633629722604007, 19.48202824504214113353894664780

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