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Properties

Label 1-3895-3895.1112-r0-0-0
Degree $1$
Conductor $3895$
Sign $0.882 - 0.469i$
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  + (−0.999 + 0.0348i)2-s + (−0.173 + 0.984i)3-s + (0.997 − 0.0697i)4-s + (0.139 − 0.990i)6-s + (−0.669 + 0.743i)7-s + (−0.994 + 0.104i)8-s + (−0.939 − 0.342i)9-s + (0.994 − 0.104i)11-s + (−0.104 + 0.994i)12-s + (−0.990 − 0.139i)13-s + (0.642 − 0.766i)14-s + (0.990 − 0.139i)16-s + (0.559 − 0.829i)17-s + (0.951 + 0.309i)18-s + (−0.615 − 0.788i)21-s + (−0.990 + 0.139i)22-s + ⋯
L(s)  = 1  + (−0.999 + 0.0348i)2-s + (−0.173 + 0.984i)3-s + (0.997 − 0.0697i)4-s + (0.139 − 0.990i)6-s + (−0.669 + 0.743i)7-s + (−0.994 + 0.104i)8-s + (−0.939 − 0.342i)9-s + (0.994 − 0.104i)11-s + (−0.104 + 0.994i)12-s + (−0.990 − 0.139i)13-s + (0.642 − 0.766i)14-s + (0.990 − 0.139i)16-s + (0.559 − 0.829i)17-s + (0.951 + 0.309i)18-s + (−0.615 − 0.788i)21-s + (−0.990 + 0.139i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3895\)    =    \(5 \cdot 19 \cdot 41\)
Sign: $0.882 - 0.469i$
Analytic conductor: \(18.0883\)
Root analytic conductor: \(18.0883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3895} (1112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3895,\ (0:\ ),\ 0.882 - 0.469i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3331840046 - 0.08307650940i\)
\(L(\frac12)\) \(\approx\) \(0.3331840046 - 0.08307650940i\)
\(L(1)\) \(\approx\) \(0.4813751189 + 0.1827111585i\)
\(L(1)\) \(\approx\) \(0.4813751189 + 0.1827111585i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.999 + 0.0348i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.669 + 0.743i)T \)
11 \( 1 + (0.994 - 0.104i)T \)
13 \( 1 + (-0.990 - 0.139i)T \)
17 \( 1 + (0.559 - 0.829i)T \)
23 \( 1 + (-0.927 + 0.374i)T \)
29 \( 1 + (-0.829 + 0.559i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
43 \( 1 + (0.529 + 0.848i)T \)
47 \( 1 + (-0.615 + 0.788i)T \)
53 \( 1 + (0.997 - 0.0697i)T \)
59 \( 1 + (0.0348 + 0.999i)T \)
61 \( 1 + (-0.848 - 0.529i)T \)
67 \( 1 + (-0.559 - 0.829i)T \)
71 \( 1 + (-0.0697 + 0.997i)T \)
73 \( 1 + (0.984 + 0.173i)T \)
79 \( 1 + (0.984 + 0.173i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.469 - 0.882i)T \)
97 \( 1 + (-0.961 - 0.275i)T \)
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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

−18.63734203904475401923206247105, −17.85474163842472535263805370318, −17.24636788566784302142183667769, −16.71989710995136131505813271255, −16.37432397378044117063171021011, −15.16292825510392736607060350521, −14.48739961156515456037737257422, −13.8544081108912175675919930406, −12.825153315327616551035469707600, −12.294357485925285119614793057910, −11.80396422287057401407333929706, −10.876836007634071260573451998094, −10.26739812047585084839414530780, −9.50671879974317387494001863343, −8.84754001279226516559695651932, −7.96078634165682135453872476974, −7.33539322129259013093796505811, −6.85650817635135623300429704290, −6.16114198462811791175326500167, −5.470135760511311663912986830995, −4.0097184576113295180772330748, −3.34522224134044227563686723603, −2.21003736435635871403508884153, −1.6978592503376652024637666804, −0.700357705144076132764314019602, 0.19448682840761018319693935661, 1.52789708365689646734435946439, 2.58585529627990326816864723406, 3.19534582288748022853196520965, 4.0053052977559661111591961582, 5.19196420857499214921584708095, 5.75462289503290195286222560216, 6.499814351764884522390592834182, 7.311455492304168595823843398601, 8.167228366956848637278759232958, 9.06134926986536424239588696197, 9.50121273551572345943580337393, 9.8399791488595180504859704957, 10.733884722814648380117486479631, 11.56464113025738178174801253547, 11.978270300671408767030754847600, 12.64438661507643441266989722268, 14.00535262764910926311312397083, 14.71145751938874441803484120572, 15.21191224646077765942076798317, 15.94600384244654591637138561268, 16.58910399819682878887425085864, 16.87998383301943133714448291137, 17.79622208995664544559349770316, 18.39124258440616878482310485905

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