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Properties

Label 2-3744-936.571-c0-0-3
Degree $2$
Conductor $3744$
Sign $0.766 + 0.642i$
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  + (0.939 − 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: $0.766 + 0.642i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.400999343\)
\(L(\frac12)\) \(\approx\) \(1.400999343\)
\(L(1)\) 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

−8.156888057027353420778107766643, −7.87019417566769634130118262350, −7.22285914761787506946329159430, −6.65065429051718129758518698864, −6.03762030655004853084229924966, −4.30822187932156393532369697701, −3.74931965196939253762283800839, −3.17176191733616277393082727845, −2.51569730872888658402410492592, −0.808699302684641340685947552365, 1.28977832532480544903708330089, 2.43566852611020266306635428573, 3.41313418946060608837177018751, 4.15326817501144025280541996273, 4.84242655649074914315248923082, 5.61475226054624891170699215476, 6.57442314402912137176064524613, 7.63532670124595439313454670479, 8.330206873672496306525923121999, 8.697078651917257831252954447144

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