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Properties

Label 1-3895-3895.1032-r0-0-0
Degree 11
Conductor 38953895
Sign 0.836−0.547i0.836 - 0.547i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.719 + 0.694i)2-s + (−0.996 + 0.0871i)3-s + (0.0348 − 0.999i)4-s + (0.656 − 0.754i)6-s + (0.933 − 0.358i)7-s + (0.669 + 0.743i)8-s + (0.984 − 0.173i)9-s + (−0.0523 + 0.998i)11-s + (0.0523 + 0.998i)12-s + (0.656 − 0.754i)13-s + (−0.422 + 0.906i)14-s + (−0.997 − 0.0697i)16-s + (0.292 + 0.956i)17-s + (−0.587 + 0.809i)18-s + (−0.898 + 0.438i)21-s + (−0.656 − 0.754i)22-s + ⋯
L(s)  = 1  + (−0.719 + 0.694i)2-s + (−0.996 + 0.0871i)3-s + (0.0348 − 0.999i)4-s + (0.656 − 0.754i)6-s + (0.933 − 0.358i)7-s + (0.669 + 0.743i)8-s + (0.984 − 0.173i)9-s + (−0.0523 + 0.998i)11-s + (0.0523 + 0.998i)12-s + (0.656 − 0.754i)13-s + (−0.422 + 0.906i)14-s + (−0.997 − 0.0697i)16-s + (0.292 + 0.956i)17-s + (−0.587 + 0.809i)18-s + (−0.898 + 0.438i)21-s + (−0.656 − 0.754i)22-s + ⋯

Functional equation

Λ(s)=(3895s/2 ΓR(s) L(s)=((0.836−0.547i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3895s/2 ΓR(s) L(s)=((0.836−0.547i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3895 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 38953895    =    5⋅19⋅415 \cdot 19 \cdot 41
Sign: 0.836−0.547i0.836 - 0.547i
Analytic conductor: 18.088318.0883
Root analytic conductor: 18.088318.0883
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3895(1032,⋅)\chi_{3895} (1032, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 3895, (0: ), 0.836−0.547i)(1,\ 3895,\ (0:\ ),\ 0.836 - 0.547i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.6550574270−0.1952804134i0.6550574270 - 0.1952804134i
L(12)L(\frac12) ≈\approx 0.6550574270−0.1952804134i0.6550574270 - 0.1952804134i
L(1)L(1) ≈\approx 0.5919052312+0.1240889783i0.5919052312 + 0.1240889783i
L(1)L(1) ≈\approx 0.5919052312+0.1240889783i0.5919052312 + 0.1240889783i

Euler product

   L(s)=∏pFp(p−s)−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+(−0.719+0.694i)T 1 + (-0.719 + 0.694i)T
3 1+(−0.996+0.0871i)T 1 + (-0.996 + 0.0871i)T
7 1+(0.933−0.358i)T 1 + (0.933 - 0.358i)T
11 1+(−0.0523+0.998i)T 1 + (-0.0523 + 0.998i)T
13 1+(0.656−0.754i)T 1 + (0.656 - 0.754i)T
17 1+(0.292+0.956i)T 1 + (0.292 + 0.956i)T
23 1+(−0.829+0.559i)T 1 + (-0.829 + 0.559i)T
29 1+(−0.956−0.292i)T 1 + (-0.956 - 0.292i)T
31 1+(0.978−0.207i)T 1 + (0.978 - 0.207i)T
37 1+(−0.951+0.309i)T 1 + (-0.951 + 0.309i)T
43 1+(−0.961−0.275i)T 1 + (-0.961 - 0.275i)T
47 1+(−0.945+0.325i)T 1 + (-0.945 + 0.325i)T
53 1+(−0.681−0.731i)T 1 + (-0.681 - 0.731i)T
59 1+(0.719−0.694i)T 1 + (0.719 - 0.694i)T
61 1+(−0.275−0.961i)T 1 + (-0.275 - 0.961i)T
67 1+(0.956+0.292i)T 1 + (0.956 + 0.292i)T
71 1+(0.681−0.731i)T 1 + (0.681 - 0.731i)T
73 1+(−0.766−0.642i)T 1 + (-0.766 - 0.642i)T
79 1+(−0.0871−0.996i)T 1 + (-0.0871 - 0.996i)T
83 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
89 1+(−0.857+0.515i)T 1 + (-0.857 + 0.515i)T
97 1+(0.798+0.601i)T 1 + (0.798 + 0.601i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−18.517781186197265740572075468171, −18.13532081275190835064676515408, −17.31808699609111718404430659594, −16.70239937249399072704663931037, −16.12341782041481375363968569619, −15.59208759562601942236421234314, −14.29323889926806786322168086180, −13.69780854093741270657394628943, −12.897100716349000839186249490811, −12.019394015532352401715001328234, −11.54980845898185502430955721937, −11.1967633585235585138277689945, −10.42978500253160028383188011775, −9.72886837610833536456178899261, −8.78809200739901576508289821692, −8.30691740854009435970268947982, −7.44342348446241168852299852978, −6.697588788147994438063028737232, −5.864376395310550444347076858, −5.03754658597808105950270809905, −4.28003695563977173706302216614, −3.45742386047376603978480535556, −2.38508714902975896663184001829, −1.54924652823204708456340886989, −0.8803830844360693643337656506, 0.35978288918396890471769427266, 1.590983138394924204172503310858, 1.784804947171300229538830314293, 3.60961406229573334968395916617, 4.48758312991710717207859658114, 5.135382407085877100205816983914, 5.780914936627730958365238260174, 6.502539395939642626671195344758, 7.24139093369921669941254144121, 7.99040114866476245061571004590, 8.3995413165380298989273361083, 9.76474140678347394331491100423, 9.9994831897365342019182098949, 10.83007993031195171288524733086, 11.32200108148760007879599069994, 12.12885668256879224949790922242, 13.00327273899914074496319999198, 13.76492672093698307921026484642, 14.69732175713840827660713042251, 15.28125468879018025786831858927, 15.74201888747314415741863815744, 16.62498153748802532123444119733, 17.32568642049802540450245026658, 17.5892175168341526781069853643, 18.168462777703290295201807872592

Graph of the ZZ-function along the critical line