Properties

Degree $2$
Conductor $1944$
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

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-i$
Motivic weight: \(0\)
Character: $\chi_{1944} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096361973\)
\(L(\frac12)\) \(\approx\) \(1.096361973\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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