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Properties

Label 1-3895-3895.1073-r0-0-0
Degree $1$
Conductor $3895$
Sign $0.705 + 0.708i$
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  + (−0.961 − 0.275i)2-s + (−0.573 − 0.819i)3-s + (0.848 + 0.529i)4-s + (0.325 + 0.945i)6-s + (−0.933 + 0.358i)7-s + (−0.669 − 0.743i)8-s + (−0.342 + 0.939i)9-s + (−0.0523 + 0.998i)11-s + (−0.0523 − 0.998i)12-s + (−0.325 − 0.945i)13-s + (0.996 − 0.0871i)14-s + (0.438 + 0.898i)16-s + (−0.681 + 0.731i)17-s + (0.587 − 0.809i)18-s + (0.829 + 0.559i)21-s + (0.325 − 0.945i)22-s + ⋯
L(s)  = 1  + (−0.961 − 0.275i)2-s + (−0.573 − 0.819i)3-s + (0.848 + 0.529i)4-s + (0.325 + 0.945i)6-s + (−0.933 + 0.358i)7-s + (−0.669 − 0.743i)8-s + (−0.342 + 0.939i)9-s + (−0.0523 + 0.998i)11-s + (−0.0523 − 0.998i)12-s + (−0.325 − 0.945i)13-s + (0.996 − 0.0871i)14-s + (0.438 + 0.898i)16-s + (−0.681 + 0.731i)17-s + (0.587 − 0.809i)18-s + (0.829 + 0.559i)21-s + (0.325 − 0.945i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $0.705 + 0.708i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (1073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ 0.705 + 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4068178497 + 0.1690877144i\)
\(L(\frac12)\) \(\approx\) \(0.4068178497 + 0.1690877144i\)
\(L(1)\) \(\approx\) \(0.4732039271 - 0.08648532110i\)
\(L(1)\) \(\approx\) \(0.4732039271 - 0.08648532110i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.961 - 0.275i)T \)
3 \( 1 + (-0.573 - 0.819i)T \)
7 \( 1 + (-0.933 + 0.358i)T \)
11 \( 1 + (-0.0523 + 0.998i)T \)
13 \( 1 + (-0.325 - 0.945i)T \)
17 \( 1 + (-0.681 + 0.731i)T \)
23 \( 1 + (0.0697 + 0.997i)T \)
29 \( 1 + (0.731 - 0.681i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (-0.241 - 0.970i)T \)
47 \( 1 + (-0.190 + 0.981i)T \)
53 \( 1 + (-0.974 + 0.224i)T \)
59 \( 1 + (-0.961 - 0.275i)T \)
61 \( 1 + (0.970 + 0.241i)T \)
67 \( 1 + (0.731 - 0.681i)T \)
71 \( 1 + (-0.974 - 0.224i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.819 + 0.573i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.0174 - 0.999i)T \)
97 \( 1 + (-0.121 + 0.992i)T \)
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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

−18.43663880167621435552080975679, −17.64099916981500797754784344230, −16.9281381620510626925287149057, −16.285599781250496220461889838729, −16.16968560611698563560816652984, −15.384480168649891000601823555725, −14.480105272465656833416947173236, −13.908875741245106103579030966113, −12.85182426504097802810803343155, −11.89336410443432911063587034605, −11.37674813185224058199779677177, −10.70539217202085072829602317015, −10.03510589840161224597428170036, −9.48060964226385759356951652009, −8.84119492052677543211831973809, −8.18669587932255312232955093778, −6.882942091874738077680041892699, −6.620869833221840600782885877090, −5.94592119012504233385692462040, −4.965986850959393592867242931620, −4.25090338135425024141857766097, −3.1266445172003805358116308620, −2.603520709306823014715445546746, −1.13120349827320923007346123584, −0.29814973198551866171216879944, 0.70980653218892957048093247310, 1.73221625098387730446528001135, 2.452798743534402850483097747063, 3.10797449744120178114224860110, 4.27865530398809904389044740342, 5.36978652361945990375395482617, 6.28111789629743097748630313037, 6.58460398096611748674684052295, 7.61596479631503574001529637873, 7.908811826392933882566319904438, 8.914110052672253336957420959182, 9.69435613691382032224158735761, 10.27266665751127174488883030456, 10.94178855873095429294607244563, 11.86475960395995742874253535985, 12.27774944096573888772788650830, 12.98337324638963907953566976637, 13.37224467178558198480531830730, 14.72214626192108290728286181523, 15.65647557921263146229167983532, 15.79230742237147917691742275932, 17.013172824259172602090025176887, 17.38016662735797060512831525806, 17.841244228455820273412226518909, 18.62568115115008100204550491738

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