↑

Properties

Label 1-3744-3744.43-r1-0-0
Degree $1$
Conductor $3744$
Sign $-0.848 - 0.529i$
Analytic cond. $402.348$
Root an. cond. $402.348$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)5-s − i·7-s + (−0.965 + 0.258i)11-s + (0.5 − 0.866i)17-s + (−0.258 − 0.965i)19-s + i·23-s + (0.866 − 0.5i)25-s + (0.258 + 0.965i)29-s + (−0.5 + 0.866i)31-s + (−0.258 − 0.965i)35-s + (0.258 − 0.965i)37-s − i·41-s + (−0.707 − 0.707i)43-s + (0.5 + 0.866i)47-s − 49-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)5-s − i·7-s + (−0.965 + 0.258i)11-s + (0.5 − 0.866i)17-s + (−0.258 − 0.965i)19-s + i·23-s + (0.866 − 0.5i)25-s + (0.258 + 0.965i)29-s + (−0.5 + 0.866i)31-s + (−0.258 − 0.965i)35-s + (0.258 − 0.965i)37-s − i·41-s + (−0.707 − 0.707i)43-s + (0.5 + 0.866i)47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.848 - 0.529i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.848 - 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1249720820 + 0.4360485574i\)
\(L(\frac12)\) \(\approx\) \(-0.1249720820 + 0.4360485574i\)
\(L(1)\) \(\approx\) \(0.7456690175 + 0.2005160159i\)
\(L(1)\) \(\approx\) \(0.7456690175 + 0.2005160159i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−18.15123713027889855788508334936, −17.11242991235855628500068497351, −16.65562968928935572650216599550, −16.13713062865530113535696194304, −15.24795728466049642412032771856, −14.74429821215939570286105651060, −13.905629087484334642547803076027, −13.07809941438100547642930210957, −12.6500188044417815703181724735, −11.76608786131571054034102919508, −11.11239018207213112870648162464, −10.2808124256890515906830530586, −10.00849339813927750290974175325, −8.61741875209662991410521748275, −8.136145676019516818086628114081, −7.63190815410930489526536775382, −6.81066805837809971832704013433, −5.935351022618694631254268951932, −5.07364588494643692020664142963, −4.16830768475136343464272909348, −3.78908548501994034283532209271, −2.86207934342460330679744000537, −1.773296105003158814955261959448, −0.70624260475143708684853099775, −0.10482876549411637871737626823, 0.98250889975566432392328032406, 2.22617043920894514785745993630, 2.91782994215593071646018087411, 3.52351875680609212591456141271, 4.72557013215863611270594346714, 5.14860515574124853651298010054, 6.006293357472208607403580531303, 7.12282176673875511670450077558, 7.4410221059287076647286124626, 8.36649205363552974853516275280, 8.96901546625293334402960333465, 9.7312416164504553336764314681, 10.69801185724375291632229011348, 11.2401789771953159666327653621, 11.98087299824545092965087686376, 12.51973302078405245155142530372, 13.228759467800053578993302360506, 14.17223871700832058439234743211, 14.8872571100108047684511055728, 15.54707268525974613954843704988, 15.89280625934213354424658235319, 16.59383763466880473720441886528, 17.76448383988789259919653655371, 18.21443742451946605853420148318, 18.75728073748977652361601268176

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