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Properties

Label 1-3871-3871.1171-r0-0-0
Degree $1$
Conductor $3871$
Sign $0.177 + 0.984i$
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  + (0.709 − 0.705i)2-s + (−0.880 + 0.474i)3-s + (0.00575 − 0.999i)4-s + (0.490 − 0.871i)5-s + (−0.289 + 0.957i)6-s + (−0.700 − 0.713i)8-s + (0.548 − 0.835i)9-s + (−0.267 − 0.963i)10-s + (−0.376 + 0.926i)11-s + (0.469 + 0.882i)12-s + (0.675 + 0.736i)13-s + (−0.0172 + 0.999i)15-s + (−0.999 − 0.0115i)16-s + (−0.778 − 0.627i)17-s + (−0.200 − 0.979i)18-s + (0.948 + 0.316i)19-s + ⋯
L(s)  = 1  + (0.709 − 0.705i)2-s + (−0.880 + 0.474i)3-s + (0.00575 − 0.999i)4-s + (0.490 − 0.871i)5-s + (−0.289 + 0.957i)6-s + (−0.700 − 0.713i)8-s + (0.548 − 0.835i)9-s + (−0.267 − 0.963i)10-s + (−0.376 + 0.926i)11-s + (0.469 + 0.882i)12-s + (0.675 + 0.736i)13-s + (−0.0172 + 0.999i)15-s + (−0.999 − 0.0115i)16-s + (−0.778 − 0.627i)17-s + (−0.200 − 0.979i)18-s + (0.948 + 0.316i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3871\)    =    \(7^{2} \cdot 79\)
Sign: $0.177 + 0.984i$
Analytic conductor: \(17.9768\)
Root analytic conductor: \(17.9768\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3871} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3871,\ (0:\ ),\ 0.177 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4441278502 + 0.3712731283i\)
\(L(\frac12)\) \(\approx\) \(0.4441278502 + 0.3712731283i\)
\(L(1)\) \(\approx\) \(0.9694721431 - 0.3444531872i\)
\(L(1)\) \(\approx\) \(0.9694721431 - 0.3444531872i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.709 - 0.705i)T \)
3 \( 1 + (-0.880 + 0.474i)T \)
5 \( 1 + (0.490 - 0.871i)T \)
11 \( 1 + (-0.376 + 0.926i)T \)
13 \( 1 + (0.675 + 0.736i)T \)
17 \( 1 + (-0.778 - 0.627i)T \)
19 \( 1 + (0.948 + 0.316i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (0.978 - 0.205i)T \)
31 \( 1 + (-0.0402 + 0.999i)T \)
37 \( 1 + (-0.991 + 0.126i)T \)
41 \( 1 + (0.509 + 0.860i)T \)
43 \( 1 + (-0.952 + 0.305i)T \)
47 \( 1 + (-0.910 - 0.413i)T \)
53 \( 1 + (-0.459 - 0.888i)T \)
59 \( 1 + (-0.684 - 0.729i)T \)
61 \( 1 + (-0.991 + 0.126i)T \)
67 \( 1 + (-0.845 + 0.534i)T \)
71 \( 1 + (-0.539 + 0.842i)T \)
73 \( 1 + (0.300 + 0.953i)T \)
83 \( 1 + (-0.928 - 0.370i)T \)
89 \( 1 + (-0.763 - 0.645i)T \)
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

−18.1187033412845041268214596635, −17.77844327949848871368302909338, −16.99952421557116323594020152964, −16.344332814726630220008094354691, −15.606802708617546132089462254781, −15.15567940554092408810592098224, −14.02940848115874370315561063926, −13.654017938098414262791780845986, −13.10628846761492863360825714233, −12.32844202896501534406443717199, −11.571381597147256860151529458090, −10.70875927342355081251082689581, −10.60368462486773537358472901309, −9.21224982391384243347210607167, −8.28594138672607660562720509141, −7.68538800250704728438170206309, −6.83780535603723985900461870784, −6.28048346537531499261210607645, −5.8062733286457706710590178941, −5.13909325470880887653839497848, −4.22939567104692905024776801431, −3.196222616282318054360007293476, −2.64724303045106225217559984670, −1.53042947911932105120614071185, −0.137001335890552233624955756710, 1.31935883526214215333382541050, 1.59937882184494480319434740204, 2.87123282449514718004100761864, 3.7861448536913702528053127212, 4.682086426814070809300934957142, 4.92399070176815207035364801685, 5.664116486640358075963074122436, 6.476973668435411315110851194700, 7.0933392375971576929284535841, 8.48693645145265591502608517560, 9.377677576343358838875545513983, 9.76742600932377017294169403910, 10.40069981854659397152694037781, 11.366938167380298398738215784999, 11.76443684516191292965059428277, 12.43387551367980818772668335456, 13.1263971088009950575661266846, 13.715070466154039756699422478628, 14.44260072253933880107222363195, 15.52472585540334831763380075208, 15.86930191943211071318555303898, 16.45898346962698567360578724211, 17.56691138872314309630372650644, 17.91254565489770140253630667793, 18.5629438310502913208595747829

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