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Properties

Label 1-3871-3871.457-r0-0-0
Degree $1$
Conductor $3871$
Sign $0.763 + 0.645i$
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  + (−0.965 + 0.261i)2-s + (0.0287 + 0.999i)3-s + (0.863 − 0.504i)4-s + (−0.999 − 0.0115i)5-s + (−0.289 − 0.957i)6-s + (−0.700 + 0.713i)8-s + (−0.998 + 0.0575i)9-s + (0.968 − 0.250i)10-s + (−0.614 + 0.788i)11-s + (0.529 + 0.848i)12-s + (0.675 − 0.736i)13-s + (−0.0172 − 0.999i)15-s + (0.490 − 0.871i)16-s + (0.932 + 0.359i)17-s + (0.948 − 0.316i)18-s + (−0.200 + 0.979i)19-s + ⋯
L(s)  = 1  + (−0.965 + 0.261i)2-s + (0.0287 + 0.999i)3-s + (0.863 − 0.504i)4-s + (−0.999 − 0.0115i)5-s + (−0.289 − 0.957i)6-s + (−0.700 + 0.713i)8-s + (−0.998 + 0.0575i)9-s + (0.968 − 0.250i)10-s + (−0.614 + 0.788i)11-s + (0.529 + 0.848i)12-s + (0.675 − 0.736i)13-s + (−0.0172 − 0.999i)15-s + (0.490 − 0.871i)16-s + (0.932 + 0.359i)17-s + (0.948 − 0.316i)18-s + (−0.200 + 0.979i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3871\)    =    \(7^{2} \cdot 79\)
Sign: $0.763 + 0.645i$
Analytic conductor: \(17.9768\)
Root analytic conductor: \(17.9768\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3871} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3871,\ (0:\ ),\ 0.763 + 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7144910851 + 0.2615475616i\)
\(L(\frac12)\) \(\approx\) \(0.7144910851 + 0.2615475616i\)
\(L(1)\) \(\approx\) \(0.5553221115 + 0.2411720682i\)
\(L(1)\) \(\approx\) \(0.5553221115 + 0.2411720682i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.965 + 0.261i)T \)
3 \( 1 + (0.0287 + 0.999i)T \)
5 \( 1 + (-0.999 - 0.0115i)T \)
11 \( 1 + (-0.614 + 0.788i)T \)
13 \( 1 + (0.675 - 0.736i)T \)
17 \( 1 + (0.932 + 0.359i)T \)
19 \( 1 + (-0.200 + 0.979i)T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (0.978 + 0.205i)T \)
31 \( 1 + (-0.845 + 0.534i)T \)
37 \( 1 + (0.605 - 0.795i)T \)
41 \( 1 + (0.509 - 0.860i)T \)
43 \( 1 + (-0.952 - 0.305i)T \)
47 \( 1 + (0.0976 - 0.995i)T \)
53 \( 1 + (0.998 - 0.0460i)T \)
59 \( 1 + (0.973 + 0.228i)T \)
61 \( 1 + (0.605 - 0.795i)T \)
67 \( 1 + (-0.0402 + 0.999i)T \)
71 \( 1 + (-0.539 - 0.842i)T \)
73 \( 1 + (-0.976 + 0.216i)T \)
83 \( 1 + (-0.928 + 0.370i)T \)
89 \( 1 + (-0.177 - 0.984i)T \)
97 \( 1 + (-0.748 - 0.663i)T \)
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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

−18.63421465381654851437295776426, −18.08771505988655342070425997239, −17.18094252684434894546611003571, −16.38099983328374237353270941345, −16.16175773665697156228714132069, −15.11994491446231582879931105761, −14.47606497188301755394968239127, −13.37428852545847321178283413846, −12.93711499186816617046772163460, −12.05448117190993277426621100189, −11.399078228030600145433039575994, −11.17768095907526583040903191154, −10.249263665194258366817467383246, −9.119539892879230726834646400599, −8.59164135352979207218877511848, −8.01740365249656425286613944652, −7.405666879869314412128910168995, −6.71062615345534353374417626384, −6.10147040494225875099831342547, −4.98246331247352849981382831424, −3.83385085276250910444238739631, −2.93191818934225560348458172968, −2.5601329626792931127074699973, −1.18800439880061033129566222621, −0.75618127009525995005152893455, 0.47200693371997050356594603617, 1.61014012640826495551331411750, 2.81514827947520507885805652485, 3.44859067196502789426943027059, 4.23030033343557677666179154538, 5.39191100445843562319655549051, 5.62619713378398322998115529486, 6.93144366203045015332844247363, 7.55087816588994376678721227192, 8.35890468743401833671837963169, 8.64606384051601148627516550401, 9.68533279370671772292099062192, 10.37136207804946557441707803905, 10.6702144221526702190658315512, 11.52861426205513915890712331141, 12.18553521697468791833303806722, 12.965829579529922179461274220737, 14.3382683325357688347048154701, 14.82384084764033187658040724357, 15.40778409979603445393267603883, 15.94485182638279211909417608156, 16.47110637470795007573451153015, 17.134452918752270646699860705725, 17.969664745854042478948622047420, 18.52554967368470739324784056122

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