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Properties

Label 1-3864-3864.2315-r0-0-0
Degree $1$
Conductor $3864$
Sign $-0.0452 - 0.998i$
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  + (0.928 − 0.371i)5-s + (0.235 − 0.971i)11-s + (0.415 − 0.909i)13-s + (−0.0475 − 0.998i)17-s + (0.0475 − 0.998i)19-s + (0.723 − 0.690i)25-s + (0.841 − 0.540i)29-s + (0.981 − 0.189i)31-s + (−0.786 + 0.618i)37-s + (−0.142 + 0.989i)41-s + (0.654 − 0.755i)43-s + (0.5 + 0.866i)47-s + (0.995 + 0.0950i)53-s + (−0.142 − 0.989i)55-s + (0.580 + 0.814i)59-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)5-s + (0.235 − 0.971i)11-s + (0.415 − 0.909i)13-s + (−0.0475 − 0.998i)17-s + (0.0475 − 0.998i)19-s + (0.723 − 0.690i)25-s + (0.841 − 0.540i)29-s + (0.981 − 0.189i)31-s + (−0.786 + 0.618i)37-s + (−0.142 + 0.989i)41-s + (0.654 − 0.755i)43-s + (0.5 + 0.866i)47-s + (0.995 + 0.0950i)53-s + (−0.142 − 0.989i)55-s + (0.580 + 0.814i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0452 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0452 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.0452 - 0.998i$
Analytic conductor: \(17.9443\)
Root analytic conductor: \(17.9443\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (2315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3864,\ (0:\ ),\ -0.0452 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.567910150 - 1.640523357i\)
\(L(\frac12)\) \(\approx\) \(1.567910150 - 1.640523357i\)
\(L(1)\) \(\approx\) \(1.281353639 - 0.4187943523i\)
\(L(1)\) \(\approx\) \(1.281353639 - 0.4187943523i\)

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 \( 1 + (0.928 - 0.371i)T \)
11 \( 1 + (0.235 - 0.971i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (-0.0475 - 0.998i)T \)
19 \( 1 + (0.0475 - 0.998i)T \)
29 \( 1 + (0.841 - 0.540i)T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + (-0.786 + 0.618i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.995 + 0.0950i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (0.327 - 0.945i)T \)
67 \( 1 + (-0.723 + 0.690i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (0.888 - 0.458i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (-0.981 - 0.189i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−18.76360808845531452126039175807, −17.86242575106267583242959738276, −17.51425053060907328232556006879, −16.77542888817747625831731806498, −16.073111156907739718471858974659, −15.19739007376617813825729490985, −14.52026026751540239611612796196, −14.0291357700723103160040634829, −13.3409995424679850604341850105, −12.45770084145401201814289936531, −12.017594032257125523174147172519, −10.98249285102008429546501887062, −10.275663612747746605425446303535, −9.880614664120456326819516151479, −8.94902604724704643567892503814, −8.42356400434010281932134350245, −7.275901376230724403965222606792, −6.73533858349750116518685486336, −6.040110596173994019933022020075, −5.34240202024101927888221009425, −4.34645102487615460177206068261, −3.72973450220409898534039974602, −2.64408205975009561001027254722, −1.84590306257274140571611164061, −1.32388505665883387974926201500, 0.68102555715886304662833825889, 1.241543804363296256737619982041, 2.63635628727811640681646840016, 2.874629908402212260077181394183, 4.09900655730321613252811078768, 4.971345785614896746550196138593, 5.58373368088702511684317672836, 6.288676550297548221300805243204, 6.96164858666285888170414759158, 8.03893613912937771558194858550, 8.65443720853316161259199872982, 9.278661979509371601625016751991, 10.055118633040030845007861178339, 10.68346998751237486982885623934, 11.491479199405641782057620118804, 12.166298248680110524232419195625, 13.095984010499291128885214969230, 13.67954954631139471609704729624, 13.93019393159261768708532316445, 15.01838462275526580740928759507, 15.76798309606819225707078460729, 16.288058035329607248048798355311, 17.16981258082589674057835950802, 17.624072142572946243985628764862, 18.26209982993503982078470814747

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