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Properties

Label 1-3895-3895.1039-r0-0-0
Degree $1$
Conductor $3895$
Sign $-0.329 + 0.943i$
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  + (0.984 + 0.173i)2-s + (−0.996 + 0.0871i)3-s + (0.939 + 0.342i)4-s + (−0.996 − 0.0871i)6-s + (−0.965 − 0.258i)7-s + (0.866 + 0.5i)8-s + (0.984 − 0.173i)9-s + (0.258 + 0.965i)11-s + (−0.965 − 0.258i)12-s + (−0.0871 + 0.996i)13-s + (−0.906 − 0.422i)14-s + (0.766 + 0.642i)16-s + (0.819 − 0.573i)17-s + 18-s + (0.984 + 0.173i)21-s + (0.0871 + 0.996i)22-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (−0.996 + 0.0871i)3-s + (0.939 + 0.342i)4-s + (−0.996 − 0.0871i)6-s + (−0.965 − 0.258i)7-s + (0.866 + 0.5i)8-s + (0.984 − 0.173i)9-s + (0.258 + 0.965i)11-s + (−0.965 − 0.258i)12-s + (−0.0871 + 0.996i)13-s + (−0.906 − 0.422i)14-s + (0.766 + 0.642i)16-s + (0.819 − 0.573i)17-s + 18-s + (0.984 + 0.173i)21-s + (0.0871 + 0.996i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $-0.329 + 0.943i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ -0.329 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129860160 + 1.591832876i\)
\(L(\frac12)\) \(\approx\) \(1.129860160 + 1.591832876i\)
\(L(1)\) \(\approx\) \(1.274112556 + 0.4397049392i\)
\(L(1)\) \(\approx\) \(1.274112556 + 0.4397049392i\)

Euler product

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

−18.42348489891695299561180919785, −17.52126648086949244728931197688, −16.764109212995452555619314480729, −16.11967224637837067874884871509, −15.77913842348929675194461362649, −14.95225705554681272779637866409, −14.10028302237421251879806616490, −13.333472310713104944074654272255, −12.78033357600207376205338911218, −12.204037021766445784960169256716, −11.647995531398399164489695900506, −10.8312187926538295614400023015, −10.19979546509961501111771399059, −9.719314468592042714840751023554, −8.38759704296688095961894172147, −7.57300632144623769606570005509, −6.69442644745564647885685976493, −6.03528539060961747692726347, −5.696334013757921730888027716052, −4.96303545072407511910163125021, −3.85086558571430011959686726279, −3.41823070518766458060504245754, −2.4890060931416627449288337452, −1.3833349963355162231896546630, −0.48021219871676943964445674292, 1.102276211715809681368935987272, 2.052217487759799533794876356312, 3.0185657633624030445516091, 4.00319913205991818277408507920, 4.4485013795247377008583296431, 5.18328210047180560575826005716, 6.22829645716331946939349355677, 6.40147847042562620505497549104, 7.27287369736384883754028081235, 7.790261078181931131946748950088, 9.335289861796386178660982509057, 9.83283360286081857886165578228, 10.55721804266191219152108620347, 11.45093665073677213982255238554, 11.94110305221007838902260335182, 12.64009671772186712047058087951, 12.99044236190694885952135967487, 14.071186890372092711322035205904, 14.49007909743139013232049290017, 15.51321740469931754367928687535, 16.03693596402500711173555041820, 16.611168934025440510050060028540, 17.02094028570522519804956417902, 17.97072938991197842085553306336, 18.689465710803662630795509128451

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