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Properties

Label 1-3724-3724.2539-r1-0-0
Degree $1$
Conductor $3724$
Sign $0.897 + 0.441i$
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  + (0.733 − 0.680i)3-s + (0.222 + 0.974i)5-s + (0.0747 − 0.997i)9-s + (−0.826 + 0.563i)11-s + (0.0747 + 0.997i)13-s + (0.826 + 0.563i)15-s + (0.988 + 0.149i)17-s + (0.988 − 0.149i)23-s + (−0.900 + 0.433i)25-s + (−0.623 − 0.781i)27-s + (0.988 + 0.149i)29-s + (0.5 − 0.866i)31-s + (−0.222 + 0.974i)33-s + (0.988 + 0.149i)37-s + (0.733 + 0.680i)39-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)3-s + (0.222 + 0.974i)5-s + (0.0747 − 0.997i)9-s + (−0.826 + 0.563i)11-s + (0.0747 + 0.997i)13-s + (0.826 + 0.563i)15-s + (0.988 + 0.149i)17-s + (0.988 − 0.149i)23-s + (−0.900 + 0.433i)25-s + (−0.623 − 0.781i)27-s + (0.988 + 0.149i)29-s + (0.5 − 0.866i)31-s + (−0.222 + 0.974i)33-s + (0.988 + 0.149i)37-s + (0.733 + 0.680i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.897 + 0.441i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (2539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ 0.897 + 0.441i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.545941396 + 0.8260771579i\)
\(L(\frac12)\) \(\approx\) \(3.545941396 + 0.8260771579i\)
\(L(1)\) \(\approx\) \(1.517137322 + 0.03991222131i\)
\(L(1)\) \(\approx\) \(1.517137322 + 0.03991222131i\)

Euler product

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

−18.49664379854038121046599907788, −17.58458813190250223875015736255, −16.9450843648112093611109371324, −16.17361158925005928128466850974, −15.74457029780880032472047062064, −15.14256604028830883870290296739, −14.1056513092898134193265127860, −13.76912032490102918603452892917, −12.8233109619629387514904291573, −12.51550675086461514452909486916, −11.35130989087472400522049875094, −10.49594542642472891077967325084, −10.10577464412420182578865598117, −9.16081882724869781277424578812, −8.717003040770484579699153959204, −7.883738097217761377261637590538, −7.54372699884688881954253570889, −6.02631903906842796617205540206, −5.413284443357279887386067358509, −4.82150012520940534597603550343, −4.03553870471176394311265360481, −2.97860920041985819716731861345, −2.67325536643963218604358147178, −1.30913742637055407891616038119, −0.618252319224426585793454167235, 0.77796174915677779645966663235, 1.726802129957173009669160172635, 2.559871594810113287792161835921, 2.97782619184488148780247251579, 3.95244993725497910482850469767, 4.82618296659356058855896570856, 6.02441207174441925130416510362, 6.47628991088139500774718470750, 7.42331771686626202006259588786, 7.66831000955693515142156454207, 8.62525531726611685672181047546, 9.52822269483029732103381563971, 9.96702513061871265489469980446, 10.89826642830491426287685026660, 11.60279371235587268574165045775, 12.43636314832202998121529953009, 13.042766084436237626905720387001, 13.82819808078065803610950508551, 14.3310424771051843882673305324, 14.93656934699073118024820023900, 15.52601379613085130144826579822, 16.45983274338556098697479094088, 17.37580838923835970758106186267, 17.96287863033557088650023001302, 18.74432802842295429780945047340

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