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Properties

Label 1-3744-3744.85-r1-0-0
Degree $1$
Conductor $3744$
Sign $0.126 + 0.991i$
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

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.126 + 0.991i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ 0.126 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3768462817 + 0.3318121343i\)
\(L(\frac12)\) \(\approx\) \(0.3768462817 + 0.3318121343i\)
\(L(1)\) \(\approx\) \(0.7921443007 - 0.1513608528i\)
\(L(1)\) \(\approx\) \(0.7921443007 - 0.1513608528i\)

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 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (-0.965 - 0.258i)T \)
67 \( 1 + (-0.965 + 0.258i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.965 - 0.258i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−18.20125988657204549423759994144, −17.893333548802829777053036219589, −16.80076712736101969426384481641, −15.906469182345361240253715301399, −15.77168834250059478508473952613, −14.94520968851561499097088175238, −14.263230536093045542336711126659, −13.533221467725666547839049943715, −12.51473177696673135786812061001, −11.88527931266849840053173955551, −11.74752222795019099606778527468, −10.66964116201190619135922772641, −9.80823995737939758265565382911, −9.229720263132562932578916663653, −8.47546651740304178320701350547, −7.68659662699120065720707957844, −6.93094663792426789753021631268, −6.40509032199356765097960477286, −5.35790315668027730232624289027, −4.65211236497502848189603012404, −3.77364730908957146597111815977, −3.09233138225356609139683644144, −2.30796380955303689565382177030, −1.22795370780871187168037520402, −0.10869640651742671802216989105, 0.71317609018403486751668295830, 1.41214526437877190121827608757, 2.84497253690647414417036132914, 3.46711426477408956378405726082, 4.2949782422934841470968885762, 4.65288746669227668403738756176, 6.14155650934008166007028850332, 6.39471957791756454180680334042, 7.46053815310831387782216457653, 8.01535802863175527519364364548, 8.69054231022184209621819577423, 9.56303737851474107108356898058, 10.27555248631903337673087296872, 11.058099438306715610129684087667, 11.73385766569614841114949926446, 12.22362008639553604798964104376, 13.334603346063981865099918229269, 13.5540309009702509337652629794, 14.567092859878499633759275780780, 15.20481213658491697432102329170, 16.02592674366193037153639571051, 16.50268286785147768520107328943, 17.03706208486082074287395863944, 17.91440942447328898456749306287, 18.71396017103332323195869466952

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