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Properties

Label 1-3744-3744.259-r1-0-0
Degree $1$
Conductor $3744$
Sign $-0.152 + 0.988i$
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 + (−0.866 + 0.5i)7-s + (−0.965 − 0.258i)11-s − 17-s + (−0.707 − 0.707i)19-s + (0.866 + 0.5i)23-s + (−0.866 + 0.5i)25-s + (0.258 − 0.965i)29-s + (−0.5 + 0.866i)31-s + (−0.707 − 0.707i)35-s + (0.707 − 0.707i)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  + (0.258 + 0.965i)5-s + (−0.866 + 0.5i)7-s + (−0.965 − 0.258i)11-s − 17-s + (−0.707 − 0.707i)19-s + (0.866 + 0.5i)23-s + (−0.866 + 0.5i)25-s + (0.258 − 0.965i)29-s + (−0.5 + 0.866i)31-s + (−0.707 − 0.707i)35-s + (0.707 − 0.707i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.152 + 0.988i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.152 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7497786126 + 0.8740096259i\)
\(L(\frac12)\) \(\approx\) \(0.7497786126 + 0.8740096259i\)
\(L(1)\) \(\approx\) \(0.8374667083 + 0.1877119492i\)
\(L(1)\) \(\approx\) \(0.8374667083 + 0.1877119492i\)

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 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.965 - 0.258i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.258 - 0.965i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
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 \( 1 + (-0.258 - 0.965i)T \)
61 \( 1 + (0.258 - 0.965i)T \)
67 \( 1 + (-0.965 + 0.258i)T \)
71 \( 1 - iT \)
73 \( 1 - iT \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.258 + 0.965i)T \)
89 \( 1 - iT \)
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

−18.30093184841567766136330491525, −17.420395454414837014423167344267, −16.82386549720138832893532662978, −16.30859115017169338210266582293, −15.63144699778787012199376641265, −14.95383238513556406424969276164, −13.96278291579612301589361856999, −13.21117598729847765173647455630, −12.864995172575234969768735395246, −12.33583299969771930152292493721, −11.24966302433128036426637303726, −10.43996006969051745359907498217, −10.00733983665755690239327724944, −8.97851931814896400821785923464, −8.69442538679581866495432255871, −7.611391581870169847103981891295, −7.01841223062790934688984152956, −6.04441082085893422400526405127, −5.50106077781405079778494603859, −4.44694076088862694430929264569, −4.10474535810808669318929864490, −2.86409391637719130130480096825, −2.205322827945855189804846686809, −1.096392783707432001540906440680, −0.287125526440886585651779742088, 0.56339748074784647350211576408, 2.08977213899509997441026835728, 2.62805862671830184363096688179, 3.21671257596225926220122771068, 4.209048905579299487231658760549, 5.16643516302822043504168061275, 6.01681020729533977306691726931, 6.50166409911486420422586672518, 7.26315479649130704639315145404, 8.00744983671083565842689467323, 9.09014058956593137482350158736, 9.42286422071081670007893475024, 10.47149647695395563527848026105, 10.888265352756564205623997489943, 11.540346514820516186555583341061, 12.661388620057658200890356456451, 13.11005864167460786611617426152, 13.71570538750321793311135871587, 14.6161287849812944106523593897, 15.334698709493198040056664060209, 15.72711220551911032948974412512, 16.45682542042885880604846669546, 17.56630818118027296061870389398, 17.86226068923211445715810039760, 18.68564051770664269595512945971

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