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Properties

Label 1-3895-3895.107-r0-0-0
Degree $1$
Conductor $3895$
Sign $0.770 + 0.637i$
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 + (0.866 + 0.5i)3-s + (−0.669 + 0.743i)4-s + (0.104 − 0.994i)6-s + (0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (0.5 + 0.866i)9-s + (−0.309 + 0.951i)11-s + (−0.951 + 0.309i)12-s + (0.994 + 0.104i)13-s + (0.5 − 0.866i)14-s + (−0.104 − 0.994i)16-s + (0.743 − 0.669i)17-s + (0.587 − 0.809i)18-s + (0.104 + 0.994i)21-s + (0.994 − 0.104i)22-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (0.866 + 0.5i)3-s + (−0.669 + 0.743i)4-s + (0.104 − 0.994i)6-s + (0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (0.5 + 0.866i)9-s + (−0.309 + 0.951i)11-s + (−0.951 + 0.309i)12-s + (0.994 + 0.104i)13-s + (0.5 − 0.866i)14-s + (−0.104 − 0.994i)16-s + (0.743 − 0.669i)17-s + (0.587 − 0.809i)18-s + (0.104 + 0.994i)21-s + (0.994 − 0.104i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ 0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.196678232 + 0.7903494199i\)
\(L(\frac12)\) \(\approx\) \(2.196678232 + 0.7903494199i\)
\(L(1)\) \(\approx\) \(1.329221941 + 0.05088815652i\)
\(L(1)\) \(\approx\) \(1.329221941 + 0.05088815652i\)

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 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.994 + 0.104i)T \)
17 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + (0.994 + 0.104i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (-0.406 - 0.913i)T \)
47 \( 1 + (0.994 + 0.104i)T \)
53 \( 1 + (-0.743 - 0.669i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.913 - 0.406i)T \)
97 \( 1 + (0.207 - 0.978i)T \)
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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

−18.49269496531670534858135384203, −17.873776751597592514586985459859, −16.90908508740586171584114022985, −16.6470097020563318840900821366, −15.60385672714085201016712527372, −15.02398132411779399699857153534, −14.43324885310501607262120431156, −13.6875835037430139294732908713, −13.362436421590331824383732761931, −12.65930146659187108450461106948, −11.2637586502932839178989010582, −10.85339306617145778740794356959, −9.88189195729155003091114471138, −9.239868697984271248806297239286, −8.32707297602970613208921963926, −8.04496051553826244324828110065, −7.46029601852284796484582679049, −6.535957615886746841635084315213, −5.97479919043584065509378513568, −5.08580465756610294070670607262, −3.97869300309018986749183448892, −3.58686827397949880634579148647, −2.34317504233231585841264677834, −1.21431658509024990205884473190, −0.82717624726170577276275992510, 1.17719801172008467248410103008, 1.91829173689265056686423135549, 2.61280059625694938742472181557, 3.35034653326084989267288259825, 4.067075674990767355274834200033, 5.00332948557412729620393059322, 5.39181917345253836552874535394, 7.02861935897646486786494748900, 7.62190276017775579547570066127, 8.49033042773383428372435726272, 8.8685584034843623994633575929, 9.58539705696647219123683474452, 10.186418710941105466391913491014, 11.050933662311375745671330549630, 11.48292685220177273915137772067, 12.58638337417039579418330440565, 12.87521979417388994773353850263, 13.87125410671351479971015990936, 14.4309674365474784404243409874, 15.13687031112522129685138378560, 15.90048244344167714796022545369, 16.52438108117525637196027682582, 17.48484667901913024913052383580, 18.20749808557008983009848015626, 18.67706607342258048687461331524

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