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Properties

Label 2-3864-3864.2981-c0-0-0
Degree $2$
Conductor $3864$
Sign $-0.694 - 0.719i$
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  + (0.281 + 0.959i)2-s + (−0.707 + 0.707i)3-s + (−0.841 + 0.540i)4-s + (−0.0994 − 0.691i)5-s + (−0.877 − 0.479i)6-s + (−0.909 − 0.415i)7-s + (−0.755 − 0.654i)8-s − 1.00i·9-s + (0.635 − 0.290i)10-s + (0.212 − 0.977i)12-s + (0.398 + 0.871i)13-s + (0.142 − 0.989i)14-s + (0.559 + 0.418i)15-s + (0.415 − 0.909i)16-s + (0.959 − 0.281i)18-s + (0.865 + 1.34i)19-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)2-s + (−0.707 + 0.707i)3-s + (−0.841 + 0.540i)4-s + (−0.0994 − 0.691i)5-s + (−0.877 − 0.479i)6-s + (−0.909 − 0.415i)7-s + (−0.755 − 0.654i)8-s − 1.00i·9-s + (0.635 − 0.290i)10-s + (0.212 − 0.977i)12-s + (0.398 + 0.871i)13-s + (0.142 − 0.989i)14-s + (0.559 + 0.418i)15-s + (0.415 − 0.909i)16-s + (0.959 − 0.281i)18-s + (0.865 + 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.694 - 0.719i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (2981, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ -0.694 - 0.719i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7897019000\)
\(L(\frac12)\) \(\approx\) \(0.7897019000\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.281 - 0.959i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (0.909 + 0.415i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
good5 \( 1 + (0.0994 + 0.691i)T + (-0.959 + 0.281i)T^{2} \)
11 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (-0.398 - 0.871i)T + (-0.654 + 0.755i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.865 - 1.34i)T + (-0.415 + 0.909i)T^{2} \)
29 \( 1 + (0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.142 + 0.989i)T^{2} \)
37 \( 1 + (-0.959 - 0.281i)T^{2} \)
41 \( 1 + (-0.959 + 0.281i)T^{2} \)
43 \( 1 + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (1.09 - 0.497i)T + (0.654 - 0.755i)T^{2} \)
61 \( 1 + (-1.06 - 0.926i)T + (0.142 + 0.989i)T^{2} \)
67 \( 1 + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.540 - 1.84i)T + (-0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (1.53 - 0.698i)T + (0.654 - 0.755i)T^{2} \)
83 \( 1 + (0.278 - 1.93i)T + (-0.959 - 0.281i)T^{2} \)
89 \( 1 + (0.142 - 0.989i)T^{2} \)
97 \( 1 + (-0.959 + 0.281i)T^{2} \)
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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

−8.815984181866047645226058542732, −8.313021191483061119107949110683, −7.19964845660361370797113031470, −6.63139551078606530837775769915, −5.89675921806410326570508112686, −5.35385802195308107424026146607, −4.29303117925669973737782388727, −4.04805402460997014491077626080, −3.07206340970680566937420872488, −1.00027928582705773788678355403, 0.57819247168291722442836066988, 1.88450833081773476069961506088, 2.97495754822258658244609118898, 3.30545417516469703968141758148, 4.65515925161014436586174179824, 5.46532723032395700541761132403, 6.02560245632853262402370084908, 6.80215456574165155305204094935, 7.53330135138715239395137096368, 8.460355871452536739964243059109

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