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Properties

Label 2-3744-416.51-c0-0-0
Degree $2$
Conductor $3744$
Sign $0.923 - 0.382i$
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  + (0.555 − 0.831i)2-s + (−0.382 − 0.923i)4-s + (0.636 + 1.53i)5-s + (−0.980 − 0.195i)8-s + (1.63 + 0.324i)10-s + (0.750 + 1.81i)11-s + (−0.923 − 0.382i)13-s + (−0.707 + 0.707i)16-s + (1.17 − 1.17i)20-s + (1.92 + 0.382i)22-s + (−1.24 + 1.24i)25-s + (−0.831 + 0.555i)26-s + (0.195 + 0.980i)32-s + (−0.324 − 1.63i)40-s + (−0.785 + 0.785i)41-s + ⋯
L(s)  = 1  + (0.555 − 0.831i)2-s + (−0.382 − 0.923i)4-s + (0.636 + 1.53i)5-s + (−0.980 − 0.195i)8-s + (1.63 + 0.324i)10-s + (0.750 + 1.81i)11-s + (−0.923 − 0.382i)13-s + (−0.707 + 0.707i)16-s + (1.17 − 1.17i)20-s + (1.92 + 0.382i)22-s + (−1.24 + 1.24i)25-s + (−0.831 + 0.555i)26-s + (0.195 + 0.980i)32-s + (−0.324 − 1.63i)40-s + (−0.785 + 0.785i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.675207259\)
\(L(\frac12)\) \(\approx\) \(1.675207259\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.555 + 0.831i)T \)
3 \( 1 \)
13 \( 1 + (0.923 + 0.382i)T \)
good5 \( 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.750 - 1.81i)T + (-0.707 + 0.707i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.785 - 0.785i)T - iT^{2} \)
43 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 - 0.390iT - T^{2} \)
53 \( 1 + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-1.02 + 0.425i)T + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 0.765T + T^{2} \)
83 \( 1 + (-1.53 - 0.636i)T + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
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

−9.235783460402157193203617729895, −7.81014184721036776818757013941, −6.98315778236426012880611243198, −6.58283390413636991512588318245, −5.69435508251344172766450686251, −4.81965603873676187007636203876, −4.06443380209001536002401461883, −3.06339013521347888308082394934, −2.38571183089444794710302306711, −1.67403356825560934344185455718, 0.796706179152182334106899950194, 2.21619113403132724087536378468, 3.47573166319672344499301491135, 4.20685600749406647246244070242, 5.14875458997272617669113774674, 5.49593161558445825749012433219, 6.30569578785809617255997815439, 6.98793786549942548095393419289, 8.178680127359539785606655986534, 8.435827181950459531231193688656

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