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Properties

Label 1-3724-3724.887-r1-0-0
Degree $1$
Conductor $3724$
Sign $-0.544 + 0.838i$
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  + (0.318 − 0.947i)3-s + (−0.980 + 0.198i)5-s + (−0.797 − 0.603i)9-s + (−0.955 + 0.294i)11-s + (0.456 + 0.889i)13-s + (−0.124 + 0.992i)15-s + (0.969 + 0.246i)17-s + (−0.270 + 0.962i)23-s + (0.921 − 0.388i)25-s + (−0.826 + 0.563i)27-s + (0.583 + 0.811i)29-s − 31-s + (−0.0249 + 0.999i)33-s + (−0.826 − 0.563i)37-s + (0.988 − 0.149i)39-s + ⋯
L(s)  = 1  + (0.318 − 0.947i)3-s + (−0.980 + 0.198i)5-s + (−0.797 − 0.603i)9-s + (−0.955 + 0.294i)11-s + (0.456 + 0.889i)13-s + (−0.124 + 0.992i)15-s + (0.969 + 0.246i)17-s + (−0.270 + 0.962i)23-s + (0.921 − 0.388i)25-s + (−0.826 + 0.563i)27-s + (0.583 + 0.811i)29-s − 31-s + (−0.0249 + 0.999i)33-s + (−0.826 − 0.563i)37-s + (0.988 − 0.149i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.544 + 0.838i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (887, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ -0.544 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3157559580 + 0.5818386102i\)
\(L(\frac12)\) \(\approx\) \(0.3157559580 + 0.5818386102i\)
\(L(1)\) \(\approx\) \(0.8685528168 - 0.1051289039i\)
\(L(1)\) \(\approx\) \(0.8685528168 - 0.1051289039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.318 - 0.947i)T \)
5 \( 1 + (-0.980 + 0.198i)T \)
11 \( 1 + (-0.955 + 0.294i)T \)
13 \( 1 + (0.456 + 0.889i)T \)
17 \( 1 + (0.969 + 0.246i)T \)
23 \( 1 + (-0.270 + 0.962i)T \)
29 \( 1 + (0.583 + 0.811i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (0.980 - 0.198i)T \)
43 \( 1 + (0.853 + 0.521i)T \)
47 \( 1 + (0.456 + 0.889i)T \)
53 \( 1 + (0.969 - 0.246i)T \)
59 \( 1 + (0.0249 - 0.999i)T \)
61 \( 1 + (0.583 + 0.811i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.995 + 0.0995i)T \)
73 \( 1 + (-0.542 - 0.840i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.733 + 0.680i)T \)
89 \( 1 + (-0.124 + 0.992i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−18.42179302108023645875862038057, −17.408971920713898921954724265112, −16.561489507564887706474493570868, −16.080786608115650145688877233883, −15.55092085745974182552380566543, −14.98160300071951134702264829751, −14.24199326257631700351478222759, −13.47091009785086705963971793812, −12.64273171149087050615082004240, −11.966002089239900533791280175046, −11.11829215079138074270084878092, −10.513868696851875662006237244447, −10.04600594536006040484626018582, −8.9861523787177592557455535189, −8.335437400549788716826956285134, −7.91149296958433994988400739242, −7.12049099431984279759441649280, −5.760894044503781815000128031070, −5.36436110372101395952553529889, −4.44105424658006476057576885735, −3.79351968990089811059923070199, −3.06466847084795604373300235265, −2.437119319745481793377256039561, −0.87207657330232001380530025460, −0.130194928940599130464244464447, 0.92147894571300289924668263330, 1.75874485309068710012159060354, 2.681260063694340099417825696645, 3.45228891631051186040371038596, 4.10859787449533587115973443115, 5.235654912825371601769035355810, 5.95717393612860375931852483182, 6.925420720426458008171882244962, 7.47115918340042265132337355238, 7.94036371713144443251302300606, 8.74602692978423537461096585020, 9.42054125334397242685079329232, 10.55325231902964206883249405206, 11.14290198844469647424432244875, 11.93446571236828160705566466827, 12.46692546862549311297858323980, 13.0499562101149667106552618678, 14.00293119353515089652479036796, 14.42546387517050805875413504437, 15.218300480752533147410794472249, 16.00203414388765912529786053595, 16.4796358922308057339008416679, 17.574088250795561484708520226033, 18.135983385366537104118587582362, 18.74655316776882288519603372890

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