↑

Properties

Label 1-3744-3744.653-r1-0-0
Degree $1$
Conductor $3744$
Sign $-0.961 + 0.273i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.961 + 0.273i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.961 + 0.273i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1648326175 - 1.181625501i\)
\(L(\frac12)\) \(\approx\) \(-0.1648326175 - 1.181625501i\)
\(L(1)\) \(\approx\) \(1.020577876 - 0.3290053999i\)
\(L(1)\) \(\approx\) \(1.020577876 - 0.3290053999i\)

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

−18.55558044427407348758205487868, −18.05879135076157195804303816561, −17.730654666623433135323328448700, −16.63553586672189694447392516491, −16.228164414494606367270675133823, −15.243017600645798724276133740771, −14.867014726661702106638850321349, −13.91517095186527836293423645406, −13.29053565588075089728683655306, −12.57824510152161392397198208901, −12.16773136833790581933465329706, −11.0950338846731193070919121116, −10.37378306465011133496840049062, −9.80678131453314585479680091611, −8.95272520700869298589905497788, −8.49780935838090204040035151159, −7.598892674869892901127240880614, −6.695375133266120872908081883293, −5.77617978629030531748238293663, −5.47788510552475567635093627178, −4.707550751425099146622759799502, −3.55271194284785095026785993297, −2.71233017596477580358895017799, −1.922433417615500239690261393391, −1.35811594939968371720184101499, 0.187802174380392646056004942871, 0.84006326216600145205371693806, 2.05521421144625773184549894898, 2.67340614307862969170669670574, 3.47164768282783426021290203609, 4.69908449450465747684289145483, 4.98732999149048856955028256135, 6.14309221457533662814999364004, 6.64504885271269875551696360664, 7.460803462148846573521779715102, 8.13001887596271736221377163110, 9.146466805936413962518577760705, 9.747075807742813520966783303332, 10.43808470194893480537147838485, 10.991084141215925833023868836878, 11.675244212299996939118421416661, 12.924877599900996173993300155362, 13.29162814432832462909735496842, 13.836970113436462779029961605006, 14.48381994864707443091008454908, 15.385719603360937631731252754928, 16.05501321449787269131816698241, 16.85969581043867476817257069623, 17.32923886634874064396236982080, 18.180219147786017296195624705

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