Properties

Label 1-3724-3724.1319-r1-0-0
Degree $1$
Conductor $3724$
Sign $-0.649 + 0.760i$
Analytic cond. $400.199$
Root an. cond. $400.199$
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  + (−0.826 + 0.563i)3-s + (0.900 + 0.433i)5-s + (0.365 − 0.930i)9-s + (0.988 + 0.149i)11-s + (0.365 + 0.930i)13-s + (−0.988 + 0.149i)15-s + (0.733 + 0.680i)17-s + (0.733 − 0.680i)23-s + (0.623 + 0.781i)25-s + (0.222 + 0.974i)27-s + (0.733 + 0.680i)29-s + (0.5 + 0.866i)31-s + (−0.900 + 0.433i)33-s + (0.733 + 0.680i)37-s + (−0.826 − 0.563i)39-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)3-s + (0.900 + 0.433i)5-s + (0.365 − 0.930i)9-s + (0.988 + 0.149i)11-s + (0.365 + 0.930i)13-s + (−0.988 + 0.149i)15-s + (0.733 + 0.680i)17-s + (0.733 − 0.680i)23-s + (0.623 + 0.781i)25-s + (0.222 + 0.974i)27-s + (0.733 + 0.680i)29-s + (0.5 + 0.866i)31-s + (−0.900 + 0.433i)33-s + (0.733 + 0.680i)37-s + (−0.826 − 0.563i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.649 + 0.760i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ -0.649 + 0.760i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.127022593 + 2.445081225i\)
\(L(\frac12)\) \(\approx\) \(1.127022593 + 2.445081225i\)
\(L(1)\) \(\approx\) \(1.065943431 + 0.5069646529i\)
\(L(1)\) \(\approx\) \(1.065943431 + 0.5069646529i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.826 + 0.563i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (0.733 + 0.680i)T \)
23 \( 1 + (0.733 - 0.680i)T \)
29 \( 1 + (0.733 + 0.680i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.733 + 0.680i)T \)
41 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (-0.826 - 0.563i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + (-0.0747 + 0.997i)T \)
61 \( 1 + (-0.955 + 0.294i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.733 + 0.680i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
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

−17.95252817318273813582672852948, −17.59798821694882944234232303021, −16.91320073843135993310727708708, −16.42873390522102143025243798854, −15.627151336922219859296791491662, −14.670596329811669799515348525600, −13.85301790833040045967054236811, −13.350040146487878994197756620431, −12.71322598101286464344082959079, −11.99093846815209992288415669315, −11.38339986688111012189875355133, −10.611176502025346025736716675742, −9.78944211226354966093197955350, −9.298205654183701671346111737013, −8.221650734111645124355595741606, −7.64897898916835444346820296647, −6.556093894491911344305895977122, −6.22365108187526154993505071314, −5.338314196881242742343088979110, −4.933907426266720292731105947598, −3.79695053684220760886632812778, −2.77323511911442314020131840032, −1.83861331564186634045298819260, −1.01003168159270225647701039633, −0.5305560543795357072335565790, 1.12446697909096164694105599703, 1.48656889174035892287342400918, 2.80814003575284219059076263324, 3.59401390692849079880560291923, 4.47448291869496146786208599346, 5.102193180941456134728612897687, 6.05641823030366643349017317620, 6.544116424822816914657009895856, 6.977340770863643734606006755872, 8.37166346360736831406347900177, 9.13182360749966456720461328683, 9.70907242680076012875625969582, 10.41099805349838254782023602485, 10.94467091243972872857333926896, 11.78422349464956814161592692751, 12.30337048971785910004084500744, 13.19508267578837449946641004263, 14.03090967204969548835503809568, 14.66611183540464481309992315682, 15.13437111714497731274522629292, 16.23351067003156894982172432253, 16.80095148677385610207637778668, 17.115286076670796836268673626797, 18.02565258361448569856004276974, 18.46685685262840038657383221412

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