Properties

Label 1-3895-3895.1112-r0-0-0
Degree 11
Conductor 38953895
Sign 0.8820.469i0.882 - 0.469i
Analytic cond. 18.088318.0883
Root an. cond. 18.088318.0883
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(3895s/2ΓR(s)L(s)=((0.8820.469i)Λ(1s)\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}
Λ(s)=(3895s/2ΓR(s)L(s)=((0.8820.469i)Λ(1s)\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: 11
Conductor: 38953895    =    519415 \cdot 19 \cdot 41
Sign: 0.8820.469i0.882 - 0.469i
Analytic conductor: 18.088318.0883
Root analytic conductor: 18.088318.0883
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3895(1112,)\chi_{3895} (1112, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 3895, (0: ), 0.8820.469i)(1,\ 3895,\ (0:\ ),\ 0.882 - 0.469i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.33318400460.08307650940i0.3331840046 - 0.08307650940i
L(12)L(\frac12) \approx 0.33318400460.08307650940i0.3331840046 - 0.08307650940i
L(1)L(1) \approx 0.4813751189+0.1827111585i0.4813751189 + 0.1827111585i
L(1)L(1) \approx 0.4813751189+0.1827111585i0.4813751189 + 0.1827111585i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
19 1 1
41 1 1
good2 1+(0.999+0.0348i)T 1 + (-0.999 + 0.0348i)T
3 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
7 1+(0.669+0.743i)T 1 + (-0.669 + 0.743i)T
11 1+(0.9940.104i)T 1 + (0.994 - 0.104i)T
13 1+(0.9900.139i)T 1 + (-0.990 - 0.139i)T
17 1+(0.5590.829i)T 1 + (0.559 - 0.829i)T
23 1+(0.927+0.374i)T 1 + (-0.927 + 0.374i)T
29 1+(0.829+0.559i)T 1 + (-0.829 + 0.559i)T
31 1+(0.9130.406i)T 1 + (-0.913 - 0.406i)T
37 1+(0.587+0.809i)T 1 + (-0.587 + 0.809i)T
43 1+(0.529+0.848i)T 1 + (0.529 + 0.848i)T
47 1+(0.615+0.788i)T 1 + (-0.615 + 0.788i)T
53 1+(0.9970.0697i)T 1 + (0.997 - 0.0697i)T
59 1+(0.0348+0.999i)T 1 + (0.0348 + 0.999i)T
61 1+(0.8480.529i)T 1 + (-0.848 - 0.529i)T
67 1+(0.5590.829i)T 1 + (-0.559 - 0.829i)T
71 1+(0.0697+0.997i)T 1 + (-0.0697 + 0.997i)T
73 1+(0.984+0.173i)T 1 + (0.984 + 0.173i)T
79 1+(0.984+0.173i)T 1 + (0.984 + 0.173i)T
83 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
89 1+(0.4690.882i)T 1 + (-0.469 - 0.882i)T
97 1+(0.9610.275i)T 1 + (-0.961 - 0.275i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line