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Properties

Label 2-3864-3864.275-c0-0-7
Degree $2$
Conductor $3864$
Sign $-0.0633 - 0.997i$
Analytic cond. $1.92838$
Root an. cond. $1.38866$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.499 + 0.866i)18-s + (−0.499 + 0.866i)21-s − 0.999·22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.499 + 0.866i)12-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.499 + 0.866i)18-s + (−0.499 + 0.866i)21-s − 0.999·22-s + (0.5 − 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6350857480\)
\(L(\frac12)\) \(\approx\) \(0.6350857480\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.219189188642780404542904303302, −7.16938022147280820033971489848, −6.50210872235113573944721055360, −5.95114725925289441576948395211, −4.94359325664082042010431851080, −4.40462312584746520102431715530, −3.11991575713817529575914785956, −2.64305245509028095759195578890, −1.31962355013442583721465561600, −0.33066867244819681763962404691, 2.22772981036653817504995239704, 3.36646395070256590935335234095, 3.93190234913587210165587762504, 5.05906572231696057694558227822, 5.27982402753625588314713162915, 6.19419868316913839343937281182, 6.74486036309056080591680124825, 7.65040822581161830897151123695, 8.504212232770323857713115624242, 9.209634382667069929620020228269

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