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Properties

Label 1-3744-3744.227-r1-0-0
Degree $1$
Conductor $3744$
Sign $0.398 + 0.917i$
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 − 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 + (0.965 − 0.258i)29-s + (−0.866 − 0.5i)31-s + (−0.965 + 0.258i)35-s + (−0.258 + 0.965i)37-s − 41-s + (0.707 − 0.707i)43-s + (−0.866 + 0.5i)47-s + 49-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)5-s − 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 + (0.965 − 0.258i)29-s + (−0.866 − 0.5i)31-s + (−0.965 + 0.258i)35-s + (−0.258 + 0.965i)37-s − 41-s + (0.707 − 0.707i)43-s + (−0.866 + 0.5i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.398 + 0.917i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ 0.398 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.165912334 + 0.7649526716i\)
\(L(\frac12)\) \(\approx\) \(1.165912334 + 0.7649526716i\)
\(L(1)\) \(\approx\) \(1.004044891 + 0.02333278709i\)
\(L(1)\) \(\approx\) \(1.004044891 + 0.02333278709i\)

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 \( 1 - T \)
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 \( 1 + (0.965 - 0.258i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 - T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
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 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 - iT \)
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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

−18.2694168968693985458401268517, −17.630707086598863652870939088943, −17.08129099158795631703314019645, −16.07638116480982248918303268887, −15.82482332521528036091937980201, −14.82986680893040222931007180640, −14.14317113726265366690204125813, −13.34159288377354323470467430168, −13.00233477182286255274570577051, −12.31194631766437604219280125782, −11.20216096532745299864588815466, −10.573166617378894951177696006594, −9.95070501794090405631676030477, −9.36953352221032294101711840311, −8.61520136434106476895571709180, −7.68655389649119335055848760371, −6.861860310480201847560221231130, −6.273049578353329816115511056079, −5.47349780431735014175515303904, −4.98361144344376511252319762248, −3.598635437031409495917313126135, −3.10255572136929393346506744296, −2.28507720346496808668773294818, −1.38730305566407877003711150812, −0.26378876084782779209711874677, 0.685968447285385739680637287247, 1.69164652839830871359849893308, 2.66899879907590216125041446570, 3.10163495253653514986143051620, 4.28750031207034959323221600319, 5.12444176354820239646830672552, 5.76848027666967986554104590078, 6.43190630398197498421234021485, 7.2132654151829319065001816041, 8.04983010257355855883808483474, 8.900713582965508797555122533491, 9.61302577882806509511417489971, 10.21057229805702887016276060156, 10.57664636614992760537510182797, 11.91024117426827828409363881762, 12.40805408806950819703259481654, 13.1517950033290842344013319511, 13.63544211667882373280830306162, 14.35195032914565052272566461195, 15.15633878873043723655055083317, 16.13138121775804134599849886830, 16.36960897277354586793314281292, 17.14937563024995512673353670800, 17.98285000011068835220368711537, 18.569334696159664441589085613982

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