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Properties

Label 1-3864-3864.1811-r0-0-0
Degree $1$
Conductor $3864$
Sign $-0.890 + 0.455i$
Analytic cond. $17.9443$
Root an. cond. $17.9443$
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  + (−0.995 + 0.0950i)5-s + (−0.327 − 0.945i)11-s + (−0.959 + 0.281i)13-s + (−0.928 − 0.371i)17-s + (0.928 − 0.371i)19-s + (0.981 − 0.189i)25-s + (−0.142 − 0.989i)29-s + (0.0475 + 0.998i)31-s + (0.580 − 0.814i)37-s + (0.415 − 0.909i)41-s + (−0.841 − 0.540i)43-s + (0.5 + 0.866i)47-s + (−0.723 + 0.690i)53-s + (0.415 + 0.909i)55-s + (0.235 − 0.971i)59-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)5-s + (−0.327 − 0.945i)11-s + (−0.959 + 0.281i)13-s + (−0.928 − 0.371i)17-s + (0.928 − 0.371i)19-s + (0.981 − 0.189i)25-s + (−0.142 − 0.989i)29-s + (0.0475 + 0.998i)31-s + (0.580 − 0.814i)37-s + (0.415 − 0.909i)41-s + (−0.841 − 0.540i)43-s + (0.5 + 0.866i)47-s + (−0.723 + 0.690i)53-s + (0.415 + 0.909i)55-s + (0.235 − 0.971i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.890 + 0.455i$
Analytic conductor: \(17.9443\)
Root analytic conductor: \(17.9443\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (1811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3864,\ (0:\ ),\ -0.890 + 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007957693377 - 0.03300122114i\)
\(L(\frac12)\) \(\approx\) \(0.007957693377 - 0.03300122114i\)
\(L(1)\) \(\approx\) \(0.7068228565 - 0.07402068578i\)
\(L(1)\) \(\approx\) \(0.7068228565 - 0.07402068578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good5 \( 1 + (-0.995 + 0.0950i)T \)
11 \( 1 + (-0.327 - 0.945i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (-0.928 - 0.371i)T \)
19 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (0.0475 + 0.998i)T \)
37 \( 1 + (0.580 - 0.814i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.723 + 0.690i)T \)
59 \( 1 + (0.235 - 0.971i)T \)
61 \( 1 + (0.888 + 0.458i)T \)
67 \( 1 + (-0.981 + 0.189i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.786 + 0.618i)T \)
79 \( 1 + (0.723 + 0.690i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (-0.0475 + 0.998i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.95964452156715448228774740241, −18.22260690685650453741129515275, −17.68646227690768196066076476928, −16.78208028166802957924818571908, −16.25363859940472850925557462258, −15.36351540965702957589103975642, −15.012874997369854979763827992055, −14.38720319519955041439532741493, −13.20253514119521106700737897579, −12.82380556262133131410412776763, −11.94892426861985061390342116233, −11.55175542558165683568618693803, −10.641599987545555036404510328787, −9.92295094833712451390910730031, −9.26123416709499599624741866259, −8.30812484366643791443808019687, −7.700197039858022478806409825956, −7.15653607000845335125747463710, −6.37159319285884505400505874574, −5.20544530124733701355002215912, −4.71493510880117275310136015858, −3.96275091416166034069360722218, −3.076535489965381629304982479430, −2.29127462539232666111616056038, −1.2554513528546650161975148908, 0.012070437311772897510700553850, 0.90627488370843643195702181077, 2.28252245525304436750938911003, 2.94778294380126192426595314108, 3.76112685971122258443628011100, 4.56934903863370085216804213300, 5.21737034366252801421240441677, 6.15615187332029504506797851385, 7.09620745521681532295205488981, 7.5131657584089380396754965008, 8.35216635436446869670978570073, 9.02357238005233854852475911684, 9.757608273651152058389336664440, 10.776837344598429888971317092620, 11.23292997897647154873453936651, 11.91997085137785733935537439600, 12.52471002307602289463939904087, 13.44556469680982280039245151770, 14.051286549784487572794749150663, 14.77145181586122908729536940039, 15.667875660299685753257299721769, 15.91842036457802128752475197546, 16.698410719244218435061396198671, 17.51700171549880991103217821562, 18.19362240908107474286930201332

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