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Properties

Label 2-3744-8.5-c1-0-12
Degree $2$
Conductor $3744$
Sign $0.559 - 0.828i$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.05i·5-s + 0.397·7-s + 0.332i·11-s + i·13-s − 4.78·17-s + 8.01i·19-s + 2.73·23-s − 4.32·25-s + 2.88i·29-s − 1.91·31-s − 1.21i·35-s + 5.85i·37-s + 2.16·41-s + 8.64i·43-s + 5.45·47-s + ⋯
L(s)  = 1  − 1.36i·5-s + 0.150·7-s + 0.100i·11-s + 0.277i·13-s − 1.16·17-s + 1.83i·19-s + 0.570·23-s − 0.864·25-s + 0.535i·29-s − 0.344·31-s − 0.205i·35-s + 0.962i·37-s + 0.338·41-s + 1.31i·43-s + 0.795·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.559 - 0.828i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ 0.559 - 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.300988315\)
\(L(\frac12)\) \(\approx\) \(1.300988315\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 + 3.05iT - 5T^{2} \)
7 \( 1 - 0.397T + 7T^{2} \)
11 \( 1 - 0.332iT - 11T^{2} \)
17 \( 1 + 4.78T + 17T^{2} \)
19 \( 1 - 8.01iT - 19T^{2} \)
23 \( 1 - 2.73T + 23T^{2} \)
29 \( 1 - 2.88iT - 29T^{2} \)
31 \( 1 + 1.91T + 31T^{2} \)
37 \( 1 - 5.85iT - 37T^{2} \)
41 \( 1 - 2.16T + 41T^{2} \)
43 \( 1 - 8.64iT - 43T^{2} \)
47 \( 1 - 5.45T + 47T^{2} \)
53 \( 1 - 9.73iT - 53T^{2} \)
59 \( 1 - 7.96iT - 59T^{2} \)
61 \( 1 + 8.09iT - 61T^{2} \)
67 \( 1 + 1.70iT - 67T^{2} \)
71 \( 1 - 5.78T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 2.71iT - 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.617291797902854244131931838267, −8.070312570489248531396141293403, −7.25393707299167372017274479389, −6.28158350914432087744713189259, −5.60901201724112469200305172975, −4.69020125298385398244928970655, −4.30215989487099154420388662576, −3.19559457222885486214787305384, −1.89368313842384863162829794134, −1.13936743611562739427445281709, 0.40333742198462400791117613555, 2.14981683515330044256645911004, 2.74065526997651325344595797641, 3.63614845249566828784557864557, 4.56720411465731974517497282535, 5.43161932776493490921571388932, 6.37845583186945838137269304928, 6.98265875874921404394091375992, 7.38326162436396789245347341080, 8.462268806738094750118233403066

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