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Properties

Label 1-3724-3724.1383-r0-0-0
Degree $1$
Conductor $3724$
Sign $0.780 - 0.625i$
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  + (−0.797 + 0.603i)3-s + (−0.797 + 0.603i)5-s + (0.270 − 0.962i)9-s + (0.900 − 0.433i)11-s + (−0.995 + 0.0995i)13-s + (0.270 − 0.962i)15-s + (0.878 − 0.478i)17-s + (0.853 − 0.521i)23-s + (0.270 − 0.962i)25-s + (0.365 + 0.930i)27-s + (−0.980 − 0.198i)29-s + (−0.5 + 0.866i)31-s + (−0.456 + 0.889i)33-s + (0.988 − 0.149i)37-s + (0.733 − 0.680i)39-s + ⋯
L(s)  = 1  + (−0.797 + 0.603i)3-s + (−0.797 + 0.603i)5-s + (0.270 − 0.962i)9-s + (0.900 − 0.433i)11-s + (−0.995 + 0.0995i)13-s + (0.270 − 0.962i)15-s + (0.878 − 0.478i)17-s + (0.853 − 0.521i)23-s + (0.270 − 0.962i)25-s + (0.365 + 0.930i)27-s + (−0.980 − 0.198i)29-s + (−0.5 + 0.866i)31-s + (−0.456 + 0.889i)33-s + (0.988 − 0.149i)37-s + (0.733 − 0.680i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.780 - 0.625i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ 0.780 - 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7416341649 - 0.2607018427i\)
\(L(\frac12)\) \(\approx\) \(0.7416341649 - 0.2607018427i\)
\(L(1)\) \(\approx\) \(0.7077989931 + 0.1039959794i\)
\(L(1)\) \(\approx\) \(0.7077989931 + 0.1039959794i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.797 + 0.603i)T \)
5 \( 1 + (-0.797 + 0.603i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (-0.995 + 0.0995i)T \)
17 \( 1 + (0.878 - 0.478i)T \)
23 \( 1 + (0.853 - 0.521i)T \)
29 \( 1 + (-0.980 - 0.198i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.988 - 0.149i)T \)
41 \( 1 + (0.124 + 0.992i)T \)
43 \( 1 + (0.998 - 0.0498i)T \)
47 \( 1 + (0.583 + 0.811i)T \)
53 \( 1 + (0.0249 + 0.999i)T \)
59 \( 1 + (-0.998 + 0.0498i)T \)
61 \( 1 + (-0.318 - 0.947i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.661 - 0.749i)T \)
73 \( 1 + (-0.411 - 0.911i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.0747 - 0.997i)T \)
89 \( 1 + (0.969 - 0.246i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−18.97973965391359628491765049465, −17.888593751075064498146505199091, −17.14567211360320662897615513371, −16.81461526510121507856945468642, −16.24322088840547849928507116661, −15.14100719760269135500596426040, −14.8050157035329200470865879153, −13.77628381067831151819908249312, −12.797409523171638894852167553875, −12.56423061555540902232413758999, −11.77193185175403561452153511452, −11.37229472252893573825933856070, −10.48313735965504363532860013686, −9.578553963842926024420549879557, −8.93494498746866594603978902684, −7.85490719575973389505125161420, −7.43353483544566790593588991420, −6.84264739360689955682045089746, −5.723320276304186289863233158819, −5.28813603237972082059248929331, −4.340886688861583428273507380217, −3.77800391371619131909231750102, −2.53940481679556443996524294868, −1.53817654799440637288381614618, −0.84380568785887187723541843026, 0.35201679587756031078018264636, 1.3634689470170704443094530892, 2.84505564271783426222954438356, 3.35667253158568668651684146854, 4.31442065287004605871815233495, 4.768328631349880227553557070933, 5.81662971256327499726084741779, 6.40713763155389344367377347314, 7.29840733509414377125099138224, 7.72485890322147248847937411092, 9.08532540339562862699398148252, 9.38025288535194640134761009049, 10.42997268797101065423849786808, 10.919372326006478594899861515618, 11.61365594682387283143769053843, 12.142605785795135707341289563558, 12.72862623365080978601303542238, 14.00224602095712280807796985625, 14.72546609954822598045908086364, 14.966872640926832437975960784975, 15.95996883804828411740408058594, 16.545156049400409741017007234550, 16.98560798071770336201296762105, 17.81705908952320298014813080837, 18.61227958307462831756671099511

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