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Properties

Label 1-3871-3871.450-r0-0-0
Degree $1$
Conductor $3871$
Sign $0.0826 - 0.996i$
Analytic cond. $17.9768$
Root an. cond. $17.9768$
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)2-s + (−0.222 − 0.974i)3-s + (0.365 − 0.930i)4-s + (−0.222 − 0.974i)5-s + (−0.733 − 0.680i)6-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (−0.733 − 0.680i)10-s + (0.0747 + 0.997i)11-s + (−0.988 − 0.149i)12-s + (0.0747 + 0.997i)13-s + (−0.900 + 0.433i)15-s + (−0.733 − 0.680i)16-s + (0.365 + 0.930i)17-s + (−0.5 + 0.866i)18-s + 19-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (−0.222 − 0.974i)3-s + (0.365 − 0.930i)4-s + (−0.222 − 0.974i)5-s + (−0.733 − 0.680i)6-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (−0.733 − 0.680i)10-s + (0.0747 + 0.997i)11-s + (−0.988 − 0.149i)12-s + (0.0747 + 0.997i)13-s + (−0.900 + 0.433i)15-s + (−0.733 − 0.680i)16-s + (0.365 + 0.930i)17-s + (−0.5 + 0.866i)18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0826 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0826 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3871\)    =    \(7^{2} \cdot 79\)
Sign: $0.0826 - 0.996i$
Analytic conductor: \(17.9768\)
Root analytic conductor: \(17.9768\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3871} (450, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3871,\ (0:\ ),\ 0.0826 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.776516960 - 1.635228421i\)
\(L(\frac12)\) \(\approx\) \(1.776516960 - 1.635228421i\)
\(L(1)\) \(\approx\) \(1.187031235 - 0.9258295657i\)
\(L(1)\) \(\approx\) \(1.187031235 - 0.9258295657i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.826 - 0.563i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (0.0747 + 0.997i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 + T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (0.365 + 0.930i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (0.955 - 0.294i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
83 \( 1 + (0.0747 - 0.997i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 + T \)
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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

−18.41243756682465160243879476294, −17.98627976700013082394603426527, −17.12171406780237695161131429772, −16.357009211975145072768502167068, −15.90837492642446082135076509835, −15.33974820370645595447583633127, −14.61487024628589234677607324960, −14.123016869712025880368921990363, −13.548536370002280372134919538331, −12.46936766551373051739038471517, −11.769593649722116133550996730622, −11.051640066261888343042811915111, −10.72944913367709106566054731143, −9.71372309275280576037385584570, −8.927848801830628923346521348356, −7.97106047040788780865726476668, −7.49615319577841834533017729683, −6.40415510328046395336095181353, −5.97176199504550313166408015037, −5.17329776913692735595991878565, −4.54436160596932253084470053584, −3.41875491617645118488395994170, −3.2157406348661411831861165925, −2.54187692384374304469975970385, −0.629777416845042453071174733741, 0.9266542059829765438826932186, 1.58116389185120709103541307678, 2.11352015302940065084915720242, 3.27924302674103773186071692040, 4.065904879568020278800597521814, 4.91610556423506970998674254347, 5.44342404827339928699126376367, 6.23917716334236869235449200996, 7.10503817824940373845575590491, 7.59885121466112925596251557163, 8.69947824282925554415879154748, 9.35871293225135409330599534284, 10.148939940599959949771118725639, 11.171024665771032123846856253275, 11.78966195856112976211133807480, 12.19321498026452151902377956685, 12.89448276518159526812648463419, 13.33527203928021599442516623396, 14.08979265813956350815816860756, 14.77664767614268428073685763347, 15.550766012372528558048933672084, 16.40125263761070492135848254171, 16.99886385193657146668812634643, 17.77007542032678595828886507548, 18.59165183390687357727187307271

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