↑

Properties

Label 1-3744-3744.211-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: $\chi_{3744} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.961 + 0.273i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1612783306 + 1.156146102i\)
\(L(\frac12)\) \(\approx\) \(0.1612783306 + 1.156146102i\)
\(L(1)\) \(\approx\) \(0.7911810793 + 0.2353337729i\)
\(L(1)\) \(\approx\) \(0.7911810793 + 0.2353337729i\)

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

−18.08845639459902530528771580004, −17.515634003705600270708896742104, −16.69049183774382846768792955645, −15.999472899972364597589831505261, −15.46292983454202370829854132139, −14.83407549681083432112193569016, −13.90989579517472027826193445384, −13.37522860423385416486638384265, −12.67274473813140320439035384334, −11.73378441669978830321072194217, −11.2334403598759625779363814946, −10.566855247496772963605290343522, −9.84498830544766486628630130837, −9.08119396627642002948647912307, −7.959324338720562464167344258807, −7.573232472495593882135289016368, −7.0954336586807424639657265538, −6.10326560794139708425191836420, −5.07342779510536541283740320803, −4.457350919520546780467815660515, −3.69314062203586111016739300634, −2.95287736710741678172249878781, −2.08253668656823840352501461730, −0.672199643869897149775773755746, −0.29488327631809271783928218488, 0.925376591354633109159514384017, 1.899925820991843498855167362399, 2.9375271807209553877805755254, 3.45302817515288000645818033986, 4.528866250819645522534137048876, 5.11810379185239040272971356432, 5.959411223651796910216272239016, 6.65274698006803303138429666984, 7.811354232926183297291738406609, 8.19228528168521524173959749268, 8.72284990093828253016828262263, 9.690435899757822058303343365374, 10.56466384663490804336702054373, 11.09178813555017456612077636959, 12.01329190153421046884782169452, 12.59894869789395138080701121733, 12.85494381400316739644232635607, 14.158397470188118363853438286, 14.71938911448530604327980404939, 15.43023767637087075901980604588, 15.96229698743102927996255000878, 16.553181948837703229639241446332, 17.32113897035816279744593229526, 18.441221822730984035342138635853, 18.66001837543347180143910497616

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