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Properties

Label 1-3895-3895.1114-r0-0-0
Degree $1$
Conductor $3895$
Sign $0.179 - 0.983i$
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  + (−0.406 − 0.913i)2-s + (0.965 − 0.258i)3-s + (−0.669 + 0.743i)4-s + (−0.629 − 0.777i)6-s + (−0.987 − 0.156i)7-s + (0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (0.891 − 0.453i)11-s + (−0.453 + 0.891i)12-s + (0.777 − 0.629i)13-s + (0.258 + 0.965i)14-s + (−0.104 − 0.994i)16-s + (0.998 + 0.0523i)17-s + (−0.809 − 0.587i)18-s + (−0.994 + 0.104i)21-s + (−0.777 − 0.629i)22-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (0.965 − 0.258i)3-s + (−0.669 + 0.743i)4-s + (−0.629 − 0.777i)6-s + (−0.987 − 0.156i)7-s + (0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (0.891 − 0.453i)11-s + (−0.453 + 0.891i)12-s + (0.777 − 0.629i)13-s + (0.258 + 0.965i)14-s + (−0.104 − 0.994i)16-s + (0.998 + 0.0523i)17-s + (−0.809 − 0.587i)18-s + (−0.994 + 0.104i)21-s + (−0.777 − 0.629i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $0.179 - 0.983i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (1114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ 0.179 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.682340011 - 1.403115613i\)
\(L(\frac12)\) \(\approx\) \(1.682340011 - 1.403115613i\)
\(L(1)\) \(\approx\) \(1.105104261 - 0.6181486091i\)
\(L(1)\) \(\approx\) \(1.105104261 - 0.6181486091i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.406 - 0.913i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-0.987 - 0.156i)T \)
11 \( 1 + (0.891 - 0.453i)T \)
13 \( 1 + (0.777 - 0.629i)T \)
17 \( 1 + (0.998 + 0.0523i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (0.998 - 0.0523i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.406 + 0.913i)T \)
47 \( 1 + (0.629 + 0.777i)T \)
53 \( 1 + (0.998 - 0.0523i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (0.406 - 0.913i)T \)
67 \( 1 + (0.0523 + 0.998i)T \)
71 \( 1 + (-0.0523 + 0.998i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.965 - 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (0.358 + 0.933i)T \)
97 \( 1 + (-0.838 + 0.544i)T \)
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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

−18.61574038463760168213526898895, −18.20507282323787822526145542794, −17.089281442647863551128411186914, −16.461679922434794340300567613474, −16.0050989084204416890041568045, −15.36270985668401749270752302482, −14.603726494860805945396344487967, −14.06621500211560343500455463029, −13.57922547567437759124577655117, −12.63182112652778764308962952588, −12.03425772639863760068966370229, −10.59928377254045718844806766494, −10.18412447101343857046167053772, −9.37447412848552531767946145923, −8.92003143726556564064269627868, −8.38044568383038488407780554707, −7.40364037597465768429588215202, −6.80166753977664782587767129245, −6.23149002676165726364767095484, −5.278674650417433258286143372759, −4.244035887525144156489076988494, −3.82105976016406883873827168815, −2.83569429324407155138999732433, −1.785345709158918627204052431537, −0.88360460716598996590924337071, 0.95176440642709606738581778317, 1.28519266054289325054174311693, 2.5902700280043887682581060818, 3.12718668639830320993656345489, 3.72085553596443665568499096163, 4.32184733312297266415851123544, 5.71243818437777302273456017337, 6.4888167626127451731182414626, 7.42396349801433861630821761297, 8.085835585941484941035002104324, 8.69918132026854015390373033450, 9.51732959869873101880412747750, 9.83673259245621815366330390645, 10.63409231322611910469893983629, 11.57584206511293567944301270760, 12.23656222963825862738816037177, 12.90356653741236077204790003001, 13.568855927291718720124220998681, 13.926492524327035626956939856926, 14.854946677160613253229548063884, 15.74543847346020937168144302322, 16.38776744386977486082174080311, 17.16851981821823598464731243697, 17.91431412207593180534297931052, 18.70208304400975257826486617954

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