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Properties

Label 1-3744-3744.1973-r1-0-0
Degree $1$
Conductor $3744$
Sign $0.861 + 0.507i$
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 + (0.866 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.861 + 0.507i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ 0.861 + 0.507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.330809121 + 0.9085592608i\)
\(L(\frac12)\) \(\approx\) \(3.330809121 + 0.9085592608i\)
\(L(1)\) \(\approx\) \(1.456388219 + 0.08395045390i\)
\(L(1)\) \(\approx\) \(1.456388219 + 0.08395045390i\)

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

−18.38509609147070044775405581330, −17.42473078619914512614809861312, −17.153842875193998367452432091630, −16.71703533903603285215086385428, −15.41571321822887774733334976304, −14.94551663545318069758591297735, −14.13337842056393866394272250174, −13.71295509664422114330087468124, −13.052895039830792927527120731233, −12.15025249352176879208033782055, −11.297648872899244744931868914742, −10.68196639834600212685947559312, −10.250842934647955539104341869392, −9.15847374865185655400145972746, −8.67971083533117949636749820565, −7.91356553956430330134475830763, −6.85990522540153778182024281198, −6.431456930492122917734505141529, −5.54795054529822613767395562642, −4.8710082583623520181646462808, −3.91705051804410219333699437518, −3.24975813404884117838906080672, −1.96139925666929569797149476902, −1.662596601882446297889822350908, −0.5795455150417800977627025062, 0.78253138293658165521704476152, 1.76157480324490567434734018299, 2.187933749328005118867031782161, 3.149236086030911190551761493801, 4.45882568471377949464008522488, 4.84134441706698126798244685521, 5.58621633551825245694263032153, 6.5691335814315832133400323985, 6.97052775868641976257522655456, 8.13889742022365814782621306237, 8.79709201857928738619955314212, 9.35604503766509476934400435335, 10.03413105523084710395114528911, 10.984837147712649395054161560603, 11.49074836826781054036295992919, 12.41280123228214687572973172962, 12.93105801402302465164212018991, 13.76638415144271591171936966148, 14.40709689366416236331390868979, 15.00969003371266746715580552081, 15.60868185472091671946849326371, 16.8254617261668584762952606009, 17.06219573278555666292134050277, 17.81107794802490605394726094740, 18.366412389428443037343161488647

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