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Properties

Label 1-3895-3895.113-r0-0-0
Degree $1$
Conductor $3895$
Sign $-0.989 - 0.141i$
Analytic cond. $18.0883$
Root an. cond. $18.0883$
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  + (−0.951 + 0.309i)2-s + i·3-s + (0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s − 9-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.951 + 0.309i)13-s − 14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (0.951 − 0.309i)18-s + (−0.309 + 0.951i)21-s + (−0.951 − 0.309i)22-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + i·3-s + (0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s − 9-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.951 + 0.309i)13-s − 14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (0.951 − 0.309i)18-s + (−0.309 + 0.951i)21-s + (−0.951 − 0.309i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $-0.989 - 0.141i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ -0.989 - 0.141i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05200015188 + 0.7333831590i\)
\(L(\frac12)\) \(\approx\) \(-0.05200015188 + 0.7333831590i\)
\(L(1)\) \(\approx\) \(0.5879420060 + 0.3908900659i\)
\(L(1)\) \(\approx\) \(0.5879420060 + 0.3908900659i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + iT \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

−18.102844232580913728221126398196, −17.60312106953656940807966764097, −16.98021861955808965358688584639, −16.65776348998138593204132193245, −15.48617008240154837098059979442, −14.61148958673430880439559754747, −14.179100819430835913308858165389, −13.26586908178788885843903993787, −12.40485351466827625005812805940, −11.85000337460914216545670790496, −11.46355767855421010826889970503, −10.482446789939241453082653175617, −9.98306289696234226777590302581, −8.88107590764575231562336367426, −8.22616879221748310117112431799, −7.931935911162578629822548198293, −7.04623284735927093486744421422, −6.43402601673676709697371863941, −5.65604071247792353172460095723, −4.557298821972133717305115410031, −3.484572006268559098501401323463, −2.73386039139824691067573756140, −1.73662286826529842894111711386, −1.34270511155987952531896116439, −0.2961505980452717329150567932, 1.15399025328717130860373637382, 2.10654986589800086457361121658, 2.80148833530968293336428480887, 3.95549978070035477193262362990, 4.87390423017895542617951433145, 5.267257663389797186880291321339, 6.27208657809822585739423757455, 7.04708876731380277995574506049, 7.92900629691840603459438793791, 8.4476202193987432997343669351, 9.30191781530897992531811780358, 9.805373159937960395287499912816, 10.28643823917971162392602565837, 11.29371400535651646722269777292, 11.810835277656462650300151902841, 12.190217090320786899869120656001, 13.918659859612870504575139139080, 14.45336516004186402924927497377, 14.80147164740296888718857177989, 15.67612932836236248651167448088, 16.1448653053870488279810116332, 16.94244688279154234104320258739, 17.5729269829017230997099212295, 17.83996097612883694930075808400, 18.90132247774901001490786083448

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