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Properties

Label 1-3895-3895.1137-r0-0-0
Degree $1$
Conductor $3895$
Sign $0.786 + 0.617i$
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  + (−0.882 + 0.469i)2-s + (−0.0871 + 0.996i)3-s + (0.559 − 0.829i)4-s + (−0.390 − 0.920i)6-s + (0.998 + 0.0523i)7-s + (−0.104 + 0.994i)8-s + (−0.984 − 0.173i)9-s + (−0.777 + 0.629i)11-s + (0.777 + 0.629i)12-s + (−0.390 − 0.920i)13-s + (−0.906 + 0.422i)14-s + (−0.374 − 0.927i)16-s + (−0.945 + 0.325i)17-s + (0.951 − 0.309i)18-s + (−0.139 + 0.990i)21-s + (0.390 − 0.920i)22-s + ⋯
L(s)  = 1  + (−0.882 + 0.469i)2-s + (−0.0871 + 0.996i)3-s + (0.559 − 0.829i)4-s + (−0.390 − 0.920i)6-s + (0.998 + 0.0523i)7-s + (−0.104 + 0.994i)8-s + (−0.984 − 0.173i)9-s + (−0.777 + 0.629i)11-s + (0.777 + 0.629i)12-s + (−0.390 − 0.920i)13-s + (−0.906 + 0.422i)14-s + (−0.374 − 0.927i)16-s + (−0.945 + 0.325i)17-s + (0.951 − 0.309i)18-s + (−0.139 + 0.990i)21-s + (0.390 − 0.920i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $0.786 + 0.617i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (1137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ 0.786 + 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7468687017 + 0.2581722831i\)
\(L(\frac12)\) \(\approx\) \(0.7468687017 + 0.2581722831i\)
\(L(1)\) \(\approx\) \(0.5952907343 + 0.2764120500i\)
\(L(1)\) \(\approx\) \(0.5952907343 + 0.2764120500i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.882 + 0.469i)T \)
3 \( 1 + (-0.0871 + 0.996i)T \)
7 \( 1 + (0.998 + 0.0523i)T \)
11 \( 1 + (-0.777 + 0.629i)T \)
13 \( 1 + (-0.390 - 0.920i)T \)
17 \( 1 + (-0.945 + 0.325i)T \)
23 \( 1 + (0.788 - 0.615i)T \)
29 \( 1 + (-0.325 + 0.945i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.0348 + 0.999i)T \)
47 \( 1 + (-0.798 - 0.601i)T \)
53 \( 1 + (-0.190 - 0.981i)T \)
59 \( 1 + (0.882 - 0.469i)T \)
61 \( 1 + (0.999 - 0.0348i)T \)
67 \( 1 + (0.325 - 0.945i)T \)
71 \( 1 + (0.190 - 0.981i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (-0.996 - 0.0871i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.974 + 0.224i)T \)
97 \( 1 + (-0.0174 - 0.999i)T \)
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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

−18.669156317811460155025418118065, −17.7756230843928040091751931879, −17.36567009369346383516263660651, −16.76405103918472757363733892022, −15.92896547298903504689391395058, −15.103515795509745599797750947547, −14.22935299389335503554484762373, −13.38916562070314391176571297445, −13.052588948790246045220363253287, −11.96965633622447761885390424646, −11.46555393313218112122266500785, −11.11691033432705894141053794510, −10.26638772781980984959296027072, −9.18582808460989274498264567623, −8.68553231110803222821522161740, −7.99015173143016190982318006538, −7.33711294354618476443452919478, −6.84663205893505058856651487787, −5.84483081914684037693971337990, −4.99743766576214701396442654137, −3.99635636653026740622263842090, −2.90003300473650720972488969456, −2.20533284496345884116240995132, −1.635425994789945082538289133121, −0.67217530696114565728601105040, 0.437662294218205416879493033453, 1.79553141260894442558306756378, 2.47275554554369897202523910631, 3.4736826570919360057870971937, 4.76184124176705836010688641228, 5.03708343179971279979440414519, 5.71395342999506944037840089087, 6.79775896350006771828998510071, 7.49971038102934923882208603796, 8.355666320219815986411818018876, 8.71393838460071051632041127765, 9.61074921534025756920916613069, 10.26173676069634650446260972491, 10.968045491822579778594244394943, 11.142337206589326899194367943654, 12.30444486281535886867544839601, 13.12225094253853240808222336384, 14.33164425824123169585941046454, 14.82604454670826119103991103013, 15.17747568056178671164996108333, 15.96282914797067162075944415167, 16.50822635135378697514517659074, 17.42224899089621256529424284348, 17.76657887450420370592364659818, 18.242787508802415606220055723388

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