Properties

Degree $2$
Conductor $3744$
Sign $0.766 - 0.642i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 + 0.342i)3-s + (0.939 + 1.62i)5-s + (0.766 − 1.32i)7-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)13-s + (0.326 + 1.85i)15-s + 0.347·17-s + (1.17 − 0.984i)21-s + (−1.26 + 2.19i)25-s + (0.500 + 0.866i)27-s + (−0.5 − 0.866i)31-s + 2.87·35-s − 1.87·37-s + (−0.173 − 0.984i)39-s + (0.173 − 0.300i)43-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)3-s + (0.939 + 1.62i)5-s + (0.766 − 1.32i)7-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)13-s + (0.326 + 1.85i)15-s + 0.347·17-s + (1.17 − 0.984i)21-s + (−1.26 + 2.19i)25-s + (0.500 + 0.866i)27-s + (−0.5 − 0.866i)31-s + 2.87·35-s − 1.87·37-s + (−0.173 − 0.984i)39-s + (0.173 − 0.300i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.766 - 0.642i$
Motivic weight: \(0\)
Character: $\chi_{3744} (3535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.295638098\)
\(L(\frac12)\) \(\approx\) \(2.295638098\)
\(L(1)\) 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 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 0.347T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.87T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.53T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.767349652447830330302247059532, −7.81105643176505738463898963639, −7.35324455429357595623416386600, −6.88186325582283943631328054631, −5.78432852953499541820823472338, −4.96712310917183544479371014632, −3.89686197769802497935337394068, −3.28881866872003456776600253792, −2.44357045637019992500120236814, −1.59438240604667677760719850812, 1.49827954681443338889068047044, 1.87183735885200913296928451193, 2.80068619899772810642715945836, 4.13290662533100953462873554894, 4.96442383802385424058142264513, 5.43364912949423072373234658277, 6.30795371785648381263106595352, 7.29019824380702548433440155366, 8.231425717918240144259201679795, 8.680908324548127657494778201423

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