Properties

Label 1-3744-3744.3379-r1-0-0
Degree $1$
Conductor $3744$
Sign $-0.854 + 0.518i$
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 + (−0.258 − 0.965i)11-s − 17-s + (0.707 + 0.707i)19-s + (−0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (0.965 − 0.258i)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 + (0.258 + 0.965i)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 + (−0.258 − 0.965i)11-s − 17-s + (0.707 + 0.707i)19-s + (−0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (0.965 − 0.258i)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 + (0.258 + 0.965i)43-s + (0.5 − 0.866i)47-s + (0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.854 + 0.518i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.854 + 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3631955085 + 1.298632204i\)
\(L(\frac12)\) \(\approx\) \(0.3631955085 + 1.298632204i\)
\(L(1)\) \(\approx\) \(1.194195063 + 0.1834266026i\)
\(L(1)\) \(\approx\) \(1.194195063 + 0.1834266026i\)

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 \( 1 + (-0.258 - 0.965i)T \)
17 \( 1 - T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.965 - 0.258i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (0.965 - 0.258i)T \)
67 \( 1 + (-0.258 + 0.965i)T \)
71 \( 1 - iT \)
73 \( 1 - iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.965 + 0.258i)T \)
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

−17.92031070343599080156367866506, −17.5618886406797716688131139403, −17.09537846041016521640940970659, −15.97894747071841625347602888162, −15.58181433266880320333146006987, −14.45451434044484394598656617542, −14.12030643230699752425334285616, −13.43523931106921969441317723628, −12.67272509952252880941222177689, −12.03701851014793679569389614900, −11.10502121424676899185704351357, −10.41685941086563362065104631583, −9.94165170367710835574366006140, −8.94372248346301762138434610761, −8.53748667355076898339021901232, −7.40718847225964518745106389113, −6.955520514316540612132559248158, −6.0621937691531492176099971471, −5.02451939249091708583366575570, −4.80682572663362329515910129626, −3.850365008908926266772299343540, −2.61882318512317481801387849593, −1.97014342034060744452801827517, −1.28217915988491211420815728771, −0.17749092692518887454017777359, 1.151197777811746564083342049939, 1.87334444156268722860824121331, 2.62299539834155991238967230830, 3.437567389215783785373423359778, 4.522958899732135422658911724430, 5.29709218996622224726795909383, 5.91848603566582111811122987434, 6.48221211666281783952381195367, 7.56213859357743373775545424140, 8.27490504186044260003306247125, 8.88567639039427845981020209342, 9.70938163649117196248422560595, 10.36562263674183212246776962161, 11.17167581690330291492923302411, 11.663855328437285321709617138984, 12.525560883756509365058203801414, 13.52026476181133034899910550747, 13.82257698009779560703717049053, 14.49572343618519457604626203244, 15.31608038307665289909701336346, 15.92930130746564028366039547075, 16.815640440393057787882840942875, 17.42127404051151107740017859149, 18.22314569630545880488365478579, 18.39465963437392663814149247884

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