Properties

Label 2-3888-9.2-c0-0-3
Degree $2$
Conductor $3888$
Sign $0.766 + 0.642i$
Analytic cond. $1.94036$
Root an. cond. $1.39296$
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  + (1.22 − 0.707i)5-s + (1.22 + 0.707i)11-s + (−0.5 − 0.866i)13-s − 1.41i·17-s + 19-s + (−1.22 + 0.707i)23-s + (0.499 − 0.866i)25-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)43-s + (0.5 + 0.866i)49-s + 1.41i·53-s + 2·55-s + (−1.22 + 0.707i)59-s + (−0.5 + 0.866i)61-s + (−1.22 − 0.707i)65-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)5-s + (1.22 + 0.707i)11-s + (−0.5 − 0.866i)13-s − 1.41i·17-s + 19-s + (−1.22 + 0.707i)23-s + (0.499 − 0.866i)25-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)43-s + (0.5 + 0.866i)49-s + 1.41i·53-s + 2·55-s + (−1.22 + 0.707i)59-s + (−0.5 + 0.866i)61-s + (−1.22 − 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3888\)    =    \(2^{4} \cdot 3^{5}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(1.94036\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3888} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3888,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.711974147\)
\(L(\frac12)\) \(\approx\) \(1.711974147\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.896737668074040109132255012847, −7.57308994276761512686756763634, −7.34049805940935753179816963115, −6.10978260143317395715461133130, −5.66207671587129000373193480172, −4.90647692300110461702931091386, −4.11269110314432836324289300077, −2.94499156292873590666922487481, −1.99952525415419847654523341150, −1.07775195513092428815528055893, 1.50130149491383411467772983689, 2.16823808429689552537651228391, 3.31047700858787553932037146888, 4.02387001413965781910743015464, 5.10026881245132127360845305760, 6.09053865758676589763139872092, 6.33496318848844507698186484945, 7.04798759738889452144865757752, 8.073956233198359884638523831097, 8.844553772283775157020330017205

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