↑

Properties

Label 1-3744-3744.1811-r0-0-0
Degree $1$
Conductor $3744$
Sign $-0.986 - 0.164i$
Analytic cond. $17.3870$
Root an. cond. $17.3870$
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  + (0.258 + 0.965i)5-s − i·7-s + (0.258 + 0.965i)11-s + (−0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s − i·23-s + (−0.866 + 0.5i)25-s + (−0.965 + 0.258i)29-s + (−0.5 + 0.866i)31-s + (0.965 − 0.258i)35-s + (0.965 + 0.258i)37-s + i·41-s + (0.707 − 0.707i)43-s + (−0.5 − 0.866i)47-s − 49-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)5-s − i·7-s + (0.258 + 0.965i)11-s + (−0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s − i·23-s + (−0.866 + 0.5i)25-s + (−0.965 + 0.258i)29-s + (−0.5 + 0.866i)31-s + (0.965 − 0.258i)35-s + (0.965 + 0.258i)37-s + i·41-s + (0.707 − 0.707i)43-s + (−0.5 − 0.866i)47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.986 - 0.164i$
Analytic conductor: \(17.3870\)
Root analytic conductor: \(17.3870\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (0:\ ),\ -0.986 - 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04011835627 + 0.4835894974i\)
\(L(\frac12)\) \(\approx\) \(-0.04011835627 + 0.4835894974i\)
\(L(1)\) \(\approx\) \(0.8689057806 + 0.2272001470i\)
\(L(1)\) \(\approx\) \(0.8689057806 + 0.2272001470i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.258 + 0.965i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.965 + 0.258i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
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 \( 1 + (0.965 + 0.258i)T \)
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 \( 1 + (-0.258 + 0.965i)T \)
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

−18.32429440040262623082511088198, −17.52636731141614541365459053720, −16.79854955773982051694358261056, −16.24255326657199573267826011365, −15.66794118098013247042350978010, −14.79787426689313949492639233168, −14.19484980114785818741818702484, −13.08490325981807612254257691394, −13.01148113670295181584025422550, −12.00082190667443982617958209695, −11.43270397841954804796972770346, −10.73582296383128585758414213888, −9.60552633990204860288399867223, −9.07316376178810803590789920451, −8.61843100627924029865400371775, −7.89538993435620518974616408047, −6.84179582348341685760881509682, −5.819540950112187267493731887421, −5.70115586197854087094743399742, −4.555768281336840862420031797359, −4.05409817194635055459569009515, −2.7259424663105321220516024820, −2.270347364887329842703168435191, −1.15933378513654067626794832206, −0.13367353163115926749510674959, 1.54390711756474290974268995103, 1.990415614881706809112990670317, 3.14919774184620616549503276897, 3.880207887593349114224372656223, 4.457304404898495662673181917779, 5.55469506692820818007046238293, 6.39143571014870462781396070318, 6.99641048770847156771987645581, 7.51844037134471992760585965369, 8.37693020442838165187502270245, 9.42636801111288155017530554504, 9.986101268162017042367963589779, 10.71256614975650398951133241883, 11.10537524771138912073635340818, 12.05253326246800007192287623511, 12.97363887827067228513730953587, 13.420013661050239117598820824899, 14.269132713235944418616059454049, 14.98737650699727569616090200883, 15.17430096331665174666229736334, 16.44320257428138913840166217780, 16.95133641277688724776538189742, 17.80943723868137951674983038178, 17.97805295175930257870997336895, 19.12026147317089893582537580136

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