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Properties

Label 2-3864-3864.1469-c0-0-5
Degree $2$
Conductor $3864$
Sign $0.0850 + 0.996i$
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  + (0.989 − 0.142i)2-s + (−0.707 − 0.707i)3-s + (0.959 − 0.281i)4-s + (1.27 − 1.47i)5-s + (−0.800 − 0.599i)6-s + (0.540 + 0.841i)7-s + (0.909 − 0.415i)8-s + 1.00i·9-s + (1.05 − 1.64i)10-s + (−0.877 − 0.479i)12-s + (−1.00 − 0.647i)13-s + (0.654 + 0.755i)14-s + (−1.94 + 0.139i)15-s + (0.841 − 0.540i)16-s + (0.142 + 0.989i)18-s + (0.562 + 1.91i)19-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)2-s + (−0.707 − 0.707i)3-s + (0.959 − 0.281i)4-s + (1.27 − 1.47i)5-s + (−0.800 − 0.599i)6-s + (0.540 + 0.841i)7-s + (0.909 − 0.415i)8-s + 1.00i·9-s + (1.05 − 1.64i)10-s + (−0.877 − 0.479i)12-s + (−1.00 − 0.647i)13-s + (0.654 + 0.755i)14-s + (−1.94 + 0.139i)15-s + (0.841 − 0.540i)16-s + (0.142 + 0.989i)18-s + (0.562 + 1.91i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0850 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0850 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.0850 + 0.996i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ 0.0850 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.555475627\)
\(L(\frac12)\) \(\approx\) \(2.555475627\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.989 + 0.142i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (-0.540 - 0.841i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
good5 \( 1 + (-1.27 + 1.47i)T + (-0.142 - 0.989i)T^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (1.00 + 0.647i)T + (0.415 + 0.909i)T^{2} \)
17 \( 1 + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.562 - 1.91i)T + (-0.841 + 0.540i)T^{2} \)
29 \( 1 + (0.841 + 0.540i)T^{2} \)
31 \( 1 + (0.654 - 0.755i)T^{2} \)
37 \( 1 + (-0.142 + 0.989i)T^{2} \)
41 \( 1 + (-0.142 - 0.989i)T^{2} \)
43 \( 1 + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (-0.0771 + 0.120i)T + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (1.28 - 0.587i)T + (0.654 - 0.755i)T^{2} \)
67 \( 1 + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (-0.281 + 0.0405i)T + (0.959 - 0.281i)T^{2} \)
73 \( 1 + (-0.841 + 0.540i)T^{2} \)
79 \( 1 + (1.03 - 1.61i)T + (-0.415 - 0.909i)T^{2} \)
83 \( 1 + (-0.627 - 0.724i)T + (-0.142 + 0.989i)T^{2} \)
89 \( 1 + (0.654 + 0.755i)T^{2} \)
97 \( 1 + (-0.142 - 0.989i)T^{2} \)
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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

−8.113324756003783343132217925604, −7.922420131153898403762010316069, −6.64965658658003376855375532151, −5.75250068934605507006546602818, −5.63954859137009577936624330743, −5.00482196527189118347388813982, −4.26420716543407216676938122673, −2.63390043299699377778847581139, −1.91948979172214567275004708664, −1.25874774407091825794388883623, 1.73990119060196406722651468546, 2.69815964516065332966589319479, 3.45825402182195360685782963281, 4.48074680769946711278432582614, 5.04601441772977697787159186831, 5.81946424369593520707779337466, 6.54063832210163758630720337954, 7.07423750435451200259485731331, 7.55089710508993660993961169283, 9.138527086834462205814548551171

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