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Properties

Label 2-3744-104.51-c0-0-1
Degree $2$
Conductor $3744$
Sign $1$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 13-s + 17-s + 2·31-s + 35-s + 37-s + 43-s + 47-s + 65-s + 71-s − 85-s + 91-s + 2·107-s + 109-s − 2·113-s − 119-s + ⋯
L(s)  = 1  − 5-s − 7-s − 13-s + 17-s + 2·31-s + 35-s + 37-s + 43-s + 47-s + 65-s + 71-s − 85-s + 91-s + 2·107-s + 109-s − 2·113-s − 119-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3744} (2287, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8360034877\)
\(L(\frac12)\) \(\approx\) \(0.8360034877\)
\(L(1)\) 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 \( 1 + T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
17 \( 1 - T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )^{2} \)
37 \( 1 - T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 - T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−8.592555844304931506315343562183, −7.81839072591238538626967763795, −7.37926231776367255841545095748, −6.51183047305905655714099644683, −5.79785836803190742644965181614, −4.78309330092551533474602983880, −4.05206224614706565562567992111, −3.21373089316726392724644421863, −2.48421811839850412848787303291, −0.76585171690408264454454696117, 0.76585171690408264454454696117, 2.48421811839850412848787303291, 3.21373089316726392724644421863, 4.05206224614706565562567992111, 4.78309330092551533474602983880, 5.79785836803190742644965181614, 6.51183047305905655714099644683, 7.37926231776367255841545095748, 7.81839072591238538626967763795, 8.592555844304931506315343562183

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