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Properties

Label 1-3895-3895.1104-r0-0-0
Degree $1$
Conductor $3895$
Sign $-0.127 + 0.991i$
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  + (−0.342 + 0.939i)2-s + (−0.573 + 0.819i)3-s + (−0.766 − 0.642i)4-s + (−0.573 − 0.819i)6-s + (0.965 − 0.258i)7-s + (0.866 − 0.5i)8-s + (−0.342 − 0.939i)9-s + (−0.258 + 0.965i)11-s + (0.965 − 0.258i)12-s + (0.819 − 0.573i)13-s + (−0.0871 + 0.996i)14-s + (0.173 + 0.984i)16-s + (0.906 − 0.422i)17-s + 18-s + (−0.342 + 0.939i)21-s + (−0.819 − 0.573i)22-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)2-s + (−0.573 + 0.819i)3-s + (−0.766 − 0.642i)4-s + (−0.573 − 0.819i)6-s + (0.965 − 0.258i)7-s + (0.866 − 0.5i)8-s + (−0.342 − 0.939i)9-s + (−0.258 + 0.965i)11-s + (0.965 − 0.258i)12-s + (0.819 − 0.573i)13-s + (−0.0871 + 0.996i)14-s + (0.173 + 0.984i)16-s + (0.906 − 0.422i)17-s + 18-s + (−0.342 + 0.939i)21-s + (−0.819 − 0.573i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $-0.127 + 0.991i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (1104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ -0.127 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9446443880 + 1.073502777i\)
\(L(\frac12)\) \(\approx\) \(0.9446443880 + 1.073502777i\)
\(L(1)\) \(\approx\) \(0.7309744864 + 0.5195160700i\)
\(L(1)\) \(\approx\) \(0.7309744864 + 0.5195160700i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.342 + 0.939i)T \)
3 \( 1 + (-0.573 + 0.819i)T \)
7 \( 1 + (0.965 - 0.258i)T \)
11 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (0.819 - 0.573i)T \)
17 \( 1 + (0.906 - 0.422i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.906 + 0.422i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (0.422 - 0.906i)T \)
53 \( 1 + (-0.0871 - 0.996i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (-0.422 + 0.906i)T \)
71 \( 1 + (-0.996 - 0.0871i)T \)
73 \( 1 + (-0.984 + 0.173i)T \)
79 \( 1 + (-0.573 + 0.819i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.819 + 0.573i)T \)
97 \( 1 + (0.906 - 0.422i)T \)
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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

−18.56776850776520816408381207072, −17.861691498876344370218064894587, −17.14258515863112232872987990177, −16.65062125499055599412088903271, −15.90011415735848500738546437794, −14.587101082657434307601536646129, −14.073470855034313711252493810017, −13.334340508214676818733635086054, −12.8035614057581927325542933801, −11.8606686720111336372035225915, −11.59405931615650263680532046640, −10.78169326756230483791977737691, −10.4832206282855192995020888196, −9.17270300526112144968913414396, −8.56502728462775306522908923691, −7.96442373946593924731102120796, −7.37923000111453738598575013917, −6.168429529689694972748046220761, −5.630470280950399879895914656972, −4.728898181495080637207488720154, −3.97743912037292396497311996887, −2.87918436927242420142864665650, −2.225856317279043212539513227291, −1.25621894962851756373535923051, −0.81749971160555952941474199401, 0.808141853015280256810744204334, 1.49987051701935414155327158138, 3.028425439773584372852057391525, 4.00614545966310265309167155389, 4.68797652436323483728688593109, 5.31187962576355995218222788257, 5.77353150711191843837651663435, 6.86714615327209640042474431838, 7.40892807507863876830123215426, 8.281558210874579375568921393130, 8.86989865327686136014751970438, 9.78939027835855986294149556457, 10.25192061293447946263266436098, 10.94969704535485239236131465847, 11.634207439811977649328459902329, 12.605721440861108537147514367920, 13.36241091480453206164497460118, 14.43128967196463685252811088569, 14.65151782328397198776980326319, 15.45764349572796045983406318788, 16.003927323620580283415984735736, 16.6292869330555580138911881655, 17.37677142009691207131979379858, 17.97514738664905977047941784940, 18.13907423974083039324331645841

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