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Properties

Label 1-3744-3744.11-r1-0-0
Degree $1$
Conductor $3744$
Sign $-0.894 + 0.446i$
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  + (−0.258 − 0.965i)5-s + (0.5 − 0.866i)7-s + (0.707 − 0.707i)11-s + (0.5 + 0.866i)17-s + (0.258 − 0.965i)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 + (−0.965 − 0.258i)35-s + (−0.258 − 0.965i)37-s + (0.5 + 0.866i)41-s + (−0.965 + 0.258i)43-s + (0.866 − 0.5i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)5-s + (0.5 − 0.866i)7-s + (0.707 − 0.707i)11-s + (0.5 + 0.866i)17-s + (0.258 − 0.965i)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 + (−0.965 − 0.258i)35-s + (−0.258 − 0.965i)37-s + (0.5 + 0.866i)41-s + (−0.965 + 0.258i)43-s + (0.866 − 0.5i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.894 + 0.446i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.894 + 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2378000280 - 1.008417568i\)
\(L(\frac12)\) \(\approx\) \(-0.2378000280 - 1.008417568i\)
\(L(1)\) \(\approx\) \(0.9739462706 - 0.3986936777i\)
\(L(1)\) \(\approx\) \(0.9739462706 - 0.3986936777i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.965 + 0.258i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.258 + 0.965i)T \)
67 \( 1 + (-0.258 + 0.965i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.258 + 0.965i)T \)
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

−18.80833363656924846395141164269, −18.3473658410026480245645364511, −17.42162494778530873780719284605, −17.01823378190266610952994257537, −15.86017096109351887114285630505, −15.32998755669994900012398604526, −14.78488694026066150682900046373, −14.168490379360269814148452548203, −13.52199550755968270124404558405, −12.29747885364272504178311037546, −11.95746937085890450059116109902, −11.3304398804875088622105164780, −10.57453538987221545364312151074, −9.61738154418881739876425294468, −9.29999190729987472500623400172, −8.101425143540957976045836765330, −7.64456522171320020346392317399, −6.830611843830148889183767237800, −6.11027900734523047778268192247, −5.33235054648129700615961576245, −4.504426876621536638161990582703, −3.59747204771515883239853703979, −2.86834362068515366662249892395, −2.063485959216529316230941760143, −1.23416248113120206588847732631, 0.15880510813588537120611279545, 1.109240083502119049443138444609, 1.44371825365575771718175108853, 2.860415090102888189256553950790, 3.75607783576695141561392592771, 4.36110962150389201023111876102, 5.08791428172717518712937617539, 5.84464943696273767219777512225, 6.84152744943144614971295717145, 7.45922541315416234178737099091, 8.41526996665735683008248826989, 8.737627941453561160153304289741, 9.5990069530858473011170341919, 10.49021373770488856994659153160, 11.189990594008988016623703954699, 11.73499823841172930304123726806, 12.63653699187230575437063389753, 13.19678625176762260878294144874, 13.88977262246523544482420438157, 14.61220228790661821282023160997, 15.26146792835102453993446756294, 16.35465900638361717843527404135, 16.55264012057304171966021555647, 17.34121174030044783961120958958, 17.78439479184086278197107937079

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