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Properties

Label 1-4032-4032.293-r0-0-0
Degree $1$
Conductor $4032$
Sign $-0.509 + 0.860i$
Analytic cond. $18.7245$
Root an. cond. $18.7245$
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  + (0.130 + 0.991i)5-s + (−0.608 − 0.793i)11-s + (−0.793 − 0.608i)13-s − i·17-s + (−0.923 + 0.382i)19-s + (0.965 + 0.258i)23-s + (−0.965 + 0.258i)25-s + (0.991 + 0.130i)29-s + (−0.5 − 0.866i)31-s + (0.923 + 0.382i)37-s + (−0.965 − 0.258i)41-s + (−0.608 − 0.793i)43-s + (0.866 + 0.5i)47-s + (−0.382 + 0.923i)53-s + (0.707 − 0.707i)55-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)5-s + (−0.608 − 0.793i)11-s + (−0.793 − 0.608i)13-s − i·17-s + (−0.923 + 0.382i)19-s + (0.965 + 0.258i)23-s + (−0.965 + 0.258i)25-s + (0.991 + 0.130i)29-s + (−0.5 − 0.866i)31-s + (0.923 + 0.382i)37-s + (−0.965 − 0.258i)41-s + (−0.608 − 0.793i)43-s + (0.866 + 0.5i)47-s + (−0.382 + 0.923i)53-s + (0.707 − 0.707i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.509 + 0.860i$
Analytic conductor: \(18.7245\)
Root analytic conductor: \(18.7245\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4032,\ (0:\ ),\ -0.509 + 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3865013786 + 0.6779526794i\)
\(L(\frac12)\) \(\approx\) \(0.3865013786 + 0.6779526794i\)
\(L(1)\) \(\approx\) \(0.8736404664 + 0.1312086445i\)
\(L(1)\) \(\approx\) \(0.8736404664 + 0.1312086445i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.130 + 0.991i)T \)
11 \( 1 + (-0.608 - 0.793i)T \)
13 \( 1 + (-0.793 - 0.608i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.923 + 0.382i)T \)
23 \( 1 + (0.965 + 0.258i)T \)
29 \( 1 + (0.991 + 0.130i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (-0.965 - 0.258i)T \)
43 \( 1 + (-0.608 - 0.793i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (0.130 + 0.991i)T \)
61 \( 1 + (-0.991 - 0.130i)T \)
67 \( 1 + (-0.608 + 0.793i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (-0.130 + 0.991i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−18.14786840718573357927738904906, −17.38694614473882234605663177368, −16.95797574030582421476123903633, −16.34248312919525564603581303055, −15.44552500563370170806220065920, −14.954148443183474213766157691120, −14.1838137053892926045082970762, −13.256723761561543424802109478585, −12.69916066650627534635615949433, −12.33626472237061179038252012517, −11.41935857983462561748111890122, −10.55402115974831661420163145321, −9.9190245918875608530792797126, −9.188863897767239004307239576734, −8.517321062719890034118423241237, −7.89387648480378542296389735192, −6.96398165683807520515955885075, −6.33949887227606748520793920850, −5.28906306942949199686687422968, −4.71588542339495855492482912748, −4.23659401306053568487458446774, −3.05376224485736912532224745135, −2.098365851805020128674660778518, −1.52542058201997344332067218531, −0.23832932046938551081705181499, 0.94843046551352096609118119974, 2.31306538730803687550965271497, 2.77844874835410277739385704841, 3.46302763516096425440398784560, 4.51321065020491003897573343488, 5.354387123046647528898339220043, 6.003467857965357072418999047472, 6.83457160092987049627835778899, 7.481149716542039830510233110924, 8.12568832771953113013925203000, 9.02938843214024815340624135944, 9.83606180440975808804831184657, 10.51899489176094843822731600865, 10.97980573231848955730096487147, 11.78030156096392518440028158182, 12.51732012431950585106400999085, 13.46454344275514841082463465437, 13.79734045160976905865519677477, 14.79418121317439006641638063236, 15.13682884220187706607264882806, 15.89039454087306532892711026902, 16.77925551312520511429203013581, 17.31035627453583993486486938396, 18.166943020161504789657820303931, 18.702607400244325984587290806397

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