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Properties

Label 2-3744-39.38-c0-0-1
Degree $2$
Conductor $3744$
Sign $-0.577 - 0.816i$
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  + 2i·7-s − 1.41·11-s + 13-s + 1.41i·17-s − 25-s − 1.41i·29-s − 1.41·47-s − 3·49-s + 1.41i·53-s − 1.41·59-s + 2i·67-s + 1.41·71-s − 2.82i·77-s + 1.41·83-s + 2i·91-s + ⋯
L(s)  = 1  + 2i·7-s − 1.41·11-s + 13-s + 1.41i·17-s − 25-s − 1.41i·29-s − 1.41·47-s − 3·49-s + 1.41i·53-s − 1.41·59-s + 2i·67-s + 1.41·71-s − 2.82i·77-s + 1.41·83-s + 2i·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3041, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9073179420\)
\(L(\frac12)\) \(\approx\) \(0.9073179420\)
\(L(1)\) 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 \( 1 \)
13 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( 1 - 2iT - T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 2iT - T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + T^{2} \)
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

−8.833239510556776321798657737592, −8.128003043767961413575405066865, −7.901784510883224051711965992428, −6.42008448299702065961655562028, −5.88497423548763766168385900960, −5.46669440232052679027970366996, −4.44749028957424214909209794350, −3.37128124558202806589129757407, −2.49891886504529678394497968118, −1.79049215104578097119088190204, 0.49703993500675149397681882914, 1.71997506190771126393735826332, 3.12747041926156170629589083624, 3.67461667684720974056528045528, 4.72516415845926324803587653670, 5.21063137483275099168203687998, 6.41820869267192493544048151116, 7.01003096028855223638669091304, 7.80199081751594944198134258925, 8.089007236226471041765851554525

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