Properties

Label 2-3744-468.295-c0-0-0
Degree $2$
Conductor $3744$
Sign $-0.941 + 0.335i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (0.5 + 0.866i)5-s − 9-s + (−0.866 + 0.5i)11-s − 13-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s i·27-s + (−0.5 − 0.866i)29-s + (−0.866 + 0.5i)31-s + (−0.5 − 0.866i)33-s + (−0.5 + 0.866i)37-s i·39-s + 2i·43-s + ⋯
L(s)  = 1  + i·3-s + (0.5 + 0.866i)5-s − 9-s + (−0.866 + 0.5i)11-s − 13-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s i·27-s + (−0.5 − 0.866i)29-s + (−0.866 + 0.5i)31-s + (−0.5 − 0.866i)33-s + (−0.5 + 0.866i)37-s i·39-s + 2i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.941 + 0.335i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.941 + 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5718186051\)
\(L(\frac12)\) \(\approx\) \(0.5718186051\)
\(L(1)\) 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 \( 1 - iT \)
13 \( 1 + T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.471691554670746466429081606600, −8.360845529085327429964977276878, −7.68278236419835727990859790227, −6.79573873328545559449639996598, −6.13795091711257480392306527663, −5.14034222876406892175774642307, −4.72160685213825204841323004927, −3.67743343580477973851153233018, −2.69376347872933197751907771488, −2.21922387356864598325314033991, 0.28781537573003314219619822609, 1.76877654736232833105143235932, 2.32945083273958573742644884867, 3.47961866012247890660246005782, 4.69362597518006149450560628997, 5.43900553299774211637137310957, 5.93030086220552215748286187389, 6.94071677654204747755156958540, 7.48783257382055484973896580869, 8.357927662309441000338273143997

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