Properties

Label 2-3744-936.259-c0-0-0
Degree $2$
Conductor $3744$
Sign $-0.939 - 0.342i$
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  + (−0.766 + 0.642i)3-s + (0.766 + 1.32i)5-s + (−0.173 + 0.300i)7-s + (0.173 − 0.984i)9-s + (0.5 + 0.866i)13-s + (−1.43 − 0.524i)15-s − 1.87·17-s + (−0.0603 − 0.342i)21-s + (−0.673 + 1.16i)25-s + (0.500 + 0.866i)27-s + (0.5 + 0.866i)31-s − 0.532·35-s − 1.53·37-s + (−0.939 − 0.342i)39-s + (−0.939 + 1.62i)43-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (0.766 + 1.32i)5-s + (−0.173 + 0.300i)7-s + (0.173 − 0.984i)9-s + (0.5 + 0.866i)13-s + (−1.43 − 0.524i)15-s − 1.87·17-s + (−0.0603 − 0.342i)21-s + (−0.673 + 1.16i)25-s + (0.500 + 0.866i)27-s + (0.5 + 0.866i)31-s − 0.532·35-s − 1.53·37-s + (−0.939 − 0.342i)39-s + (−0.939 + 1.62i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8389989341\)
\(L(\frac12)\) \(\approx\) \(0.8389989341\)
\(L(1)\) 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 \( 1 + (0.766 - 0.642i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.87T + T^{2} \)
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 \( 1 + 1.53T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 0.347T + T^{2} \)
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

−9.105043348449738596218224966998, −8.575989923248108209268106262906, −7.09871503867140076841523137529, −6.61112209766942266341281516921, −6.26238637999742559816300630318, −5.34690688900200407730970040651, −4.49719197996130716771057535405, −3.63189400667665096835793635687, −2.71691718403907785979194125584, −1.75821262403368109136986224545, 0.51500662070888455246833975896, 1.58888006257969945872339103854, 2.42675058333601504373646258366, 3.94573534024520707209492164041, 4.80563556259072697712192779817, 5.38934964461652908198626855329, 6.09393801026668791676544166501, 6.74174482189304063248025002344, 7.58533286334372654510174377935, 8.545852957948439866526079295368

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