Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{5} $
Sign $i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·5-s − 1.41i·11-s + 13-s + 1.41i·17-s − 19-s − 1.41i·23-s − 1.00·25-s − 31-s + 43-s − 49-s − 1.41i·53-s − 2.00·55-s − 1.41i·59-s + 61-s − 1.41i·65-s + ⋯
L(s)  = 1  − 1.41i·5-s − 1.41i·11-s + 13-s + 1.41i·17-s − 19-s − 1.41i·23-s − 1.00·25-s − 31-s + 43-s − 49-s − 1.41i·53-s − 2.00·55-s − 1.41i·59-s + 61-s − 1.41i·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1944\)    =    \(2^{3} \cdot 3^{5}\)
\( \varepsilon \)  =  $i$
motivic weight  =  \(0\)
character  :  $\chi_{1944} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1944,\ (\ :0),\ i)$
$L(\frac{1}{2})$  $\approx$  $1.096361973$
$L(\frac12)$  $\approx$  $1.096361973$
$L(1)$   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 \( 1 + 1.41iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−8.800539897295006893726190109579, −8.519543626831669197298280336708, −8.078500867031149834205514057112, −6.55623339616343309362604077027, −5.99662915432129118686496093463, −5.19582091798717778783100990677, −4.18809200065074855229515401565, −3.54198703495389148907683921513, −2.00262217531102882676359375349, −0.826746484795851843203846911133, 1.80286987544936973970745573275, 2.77944343197474956611119945110, 3.67935020901390904550870884872, 4.62147982372416173750603605302, 5.72350307363656571052338342490, 6.54547058168710754302432808774, 7.28209321333657416534071415815, 7.66356395948192751359859249137, 8.987129602501258249038015090450, 9.573243202668337204861428860891

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