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Properties

Label 1-3724-3724.1019-r0-0-0
Degree $1$
Conductor $3724$
Sign $0.363 - 0.931i$
Analytic cond. $17.2941$
Root an. cond. $17.2941$
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  + (0.623 + 0.781i)3-s + (0.365 − 0.930i)5-s + (−0.222 + 0.974i)9-s + (0.733 − 0.680i)11-s + (0.733 − 0.680i)13-s + (0.955 − 0.294i)15-s + (−0.900 − 0.433i)17-s + (0.900 − 0.433i)23-s + (−0.733 − 0.680i)25-s + (−0.900 + 0.433i)27-s + (−0.0747 − 0.997i)29-s + (−0.5 − 0.866i)31-s + (0.988 + 0.149i)33-s + (−0.826 + 0.563i)37-s + (0.988 + 0.149i)39-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)3-s + (0.365 − 0.930i)5-s + (−0.222 + 0.974i)9-s + (0.733 − 0.680i)11-s + (0.733 − 0.680i)13-s + (0.955 − 0.294i)15-s + (−0.900 − 0.433i)17-s + (0.900 − 0.433i)23-s + (−0.733 − 0.680i)25-s + (−0.900 + 0.433i)27-s + (−0.0747 − 0.997i)29-s + (−0.5 − 0.866i)31-s + (0.988 + 0.149i)33-s + (−0.826 + 0.563i)37-s + (0.988 + 0.149i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.363 - 0.931i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ 0.363 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.793877026 - 1.225307283i\)
\(L(\frac12)\) \(\approx\) \(1.793877026 - 1.225307283i\)
\(L(1)\) \(\approx\) \(1.383194959 - 0.1421425891i\)
\(L(1)\) \(\approx\) \(1.383194959 - 0.1421425891i\)

Euler product

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

−18.73992866953259147269148049905, −17.95023800426268018445150040212, −17.79485373197637892486506086496, −16.86393682948193653923241696312, −15.912434373634912824745191032902, −14.965519178095240537908579854, −14.69533537410219911631196095194, −13.93123541764013706367679211969, −13.369051344128074571776405221687, −12.67496494589102766901681907224, −11.87488465241432883898197478869, −11.127606750121393995297363355799, −10.52820088482586303732430934624, −9.37702516253672370348014110134, −9.08650003802266427635880613905, −8.24648401398032315207953058979, −7.15844020733401165318681202959, −6.88119520253694738303023195258, −6.30773147289221679208741481904, −5.340084093933407445861072179210, −4.11194545428941087834521913202, −3.50916977956971302797418420015, −2.6751533512888967267782237573, −1.80314248721851096415655139628, −1.34994685641948113000130104374, 0.539296000039847244707743814208, 1.57605396487877003080836910500, 2.51546392789322742748443965541, 3.34788560978971538617789788631, 4.17341428310600902945905104946, 4.71886526901070312788174450554, 5.64156825942091432976554565859, 6.18505463150512295995214814865, 7.36646638791132912698433986413, 8.24986606302479178180302028681, 8.835681472635553000690111615157, 9.20701322634901296238396482870, 10.00244879391749989461008498907, 10.86855242927337278784287154093, 11.37727227411973071520893298734, 12.36708324122304987420012207085, 13.31550513398560540271819145212, 13.5444203124641029934046637543, 14.34843473013389516750027925338, 15.19146663915454785392602427423, 15.79436661663020218487428957998, 16.32993229139390253055421827394, 17.12690859474995932388926769558, 17.504929090769266655232054588041, 18.70628920634487636927839764130

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