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Properties

Label 1-3744-3744.133-r0-0-0
Degree $1$
Conductor $3744$
Sign $0.726 - 0.686i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)5-s + (−0.866 − 0.5i)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 + (−0.5 + 0.866i)31-s + (0.965 + 0.258i)35-s + (−0.965 + 0.258i)37-s + (−0.866 + 0.5i)41-s + (0.965 − 0.258i)43-s + (0.5 + 0.866i)47-s + (0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)5-s + (−0.866 − 0.5i)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 + (−0.5 + 0.866i)31-s + (0.965 + 0.258i)35-s + (−0.965 + 0.258i)37-s + (−0.866 + 0.5i)41-s + (0.965 − 0.258i)43-s + (0.5 + 0.866i)47-s + (0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.726 - 0.686i$
Analytic conductor: \(17.3870\)
Root analytic conductor: \(17.3870\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (0:\ ),\ 0.726 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4857750300 - 0.1932280663i\)
\(L(\frac12)\) \(\approx\) \(0.4857750300 - 0.1932280663i\)
\(L(1)\) \(\approx\) \(0.6412697910 + 0.003449299154i\)
\(L(1)\) \(\approx\) \(0.6412697910 + 0.003449299154i\)

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 \( 1 + (-0.866 - 0.5i)T \)
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 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.965 - 0.258i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.258 - 0.965i)T \)
67 \( 1 + (0.965 + 0.258i)T \)
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 \( 1 + (-0.5 + 0.866i)T \)
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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

−18.73574531700625495174424136240, −18.28623387236309736671474599856, −17.15777318605785570765626225396, −16.59321337032346052260681599564, −15.78386880964551372683591728822, −15.49420063294134573949052408404, −14.76209238200449872926550390504, −13.87959423346589874751280880632, −12.92683869773286810620483379001, −12.51326895649823957887429896273, −11.941207282244239213932943514591, −11.15254358227758837825732032255, −10.26673904290703830730316824486, −9.673513897156225335714264099675, −8.86046315265544714775970579442, −8.146787473092719041984679091408, −7.435025526045711686965869628791, −6.80891832650652899239833880946, −5.819756557981227134716144014143, −5.17179087140417914019496408743, −4.19623870124419662681078989017, −3.66670471729535576975730386058, −2.64950598739328203209156086369, −2.01485068761955094624005946218, −0.50213035632043217663008231408, 0.29025415926689069810087088094, 1.52599844041185764525752663573, 2.69713690226772489480155185580, 3.55100698563734668274318057990, 3.815634007956790335232027586648, 4.90804353339041748542424603466, 5.80355180931133549206427436243, 6.57262214040426694282378444607, 7.24165932926353587970391980736, 8.021778428745446599102681369918, 8.55127976780084184818882845498, 9.48190698960004995186089006129, 10.48865974366886849195909747781, 10.705429932214393520018284601069, 11.54820571053422788147065992264, 12.60725840279288984141837914511, 12.733601061137390392470243728507, 13.78659824858475576603852871953, 14.382975813491638780896959632544, 15.30536502754099601383728203285, 15.77494362676115281493197640766, 16.43203571635362178049974369709, 16.993083574201731762446814756741, 17.90827172797103777182166223137, 18.79678991514128946105260464910

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