↑

Properties

Label 1-1441-1441.76-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.834 + 0.551i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.215 − 0.976i)2-s + (−0.998 + 0.0483i)3-s + (−0.906 − 0.421i)4-s + (0.836 + 0.548i)5-s + (−0.168 + 0.985i)6-s + (0.443 − 0.896i)7-s + (−0.607 + 0.794i)8-s + (0.995 − 0.0965i)9-s + (0.715 − 0.698i)10-s + (0.926 + 0.377i)12-s + (−0.715 − 0.698i)13-s + (−0.779 − 0.626i)14-s + (−0.861 − 0.506i)15-s + (0.644 + 0.764i)16-s + (−0.0724 − 0.997i)17-s + (0.120 − 0.992i)18-s + ⋯
L(s)  = 1  + (0.215 − 0.976i)2-s + (−0.998 + 0.0483i)3-s + (−0.906 − 0.421i)4-s + (0.836 + 0.548i)5-s + (−0.168 + 0.985i)6-s + (0.443 − 0.896i)7-s + (−0.607 + 0.794i)8-s + (0.995 − 0.0965i)9-s + (0.715 − 0.698i)10-s + (0.926 + 0.377i)12-s + (−0.715 − 0.698i)13-s + (−0.779 − 0.626i)14-s + (−0.861 − 0.506i)15-s + (0.644 + 0.764i)16-s + (−0.0724 − 0.997i)17-s + (0.120 − 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.834 + 0.551i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.834 + 0.551i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1642096020 - 0.5464289984i\)
\(L(\frac12)\) \(\approx\) \(-0.1642096020 - 0.5464289984i\)
\(L(1)\) \(\approx\) \(0.6134923840 - 0.4684958975i\)
\(L(1)\) \(\approx\) \(0.6134923840 - 0.4684958975i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.215 - 0.976i)T \)
3 \( 1 + (-0.998 + 0.0483i)T \)
5 \( 1 + (0.836 + 0.548i)T \)
7 \( 1 + (0.443 - 0.896i)T \)
13 \( 1 + (-0.715 - 0.698i)T \)
17 \( 1 + (-0.0724 - 0.997i)T \)
19 \( 1 + (-0.970 + 0.239i)T \)
23 \( 1 + (-0.958 - 0.285i)T \)
29 \( 1 + (0.995 + 0.0965i)T \)
31 \( 1 + (-0.0241 + 0.999i)T \)
37 \( 1 + (-0.485 - 0.873i)T \)
41 \( 1 + (-0.644 + 0.764i)T \)
43 \( 1 + (0.262 - 0.964i)T \)
47 \( 1 + (0.748 - 0.663i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.527 + 0.849i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.262 + 0.964i)T \)
71 \( 1 + (-0.568 - 0.822i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (-0.681 - 0.732i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.989 + 0.144i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.61928632890862518915545998538, −20.86800120921981886898092105318, −19.34037372731465103749132481629, −18.62144638200927361161264185164, −17.826169283436545515813833078343, −17.213638684420722436341405102336, −16.90812965338982509836633305559, −15.900903185358375230035009310113, −15.29840264495069308040373320492, −14.40542929652599708874574511027, −13.628240096443622746728559019523, −12.61510338664000621295709080830, −12.361893675118253071545130636820, −11.37933461014164984008598723124, −10.170252143958881128599595556410, −9.53632932648479465515516137920, −8.628031198813441112100606699757, −7.92924201816673751571198640294, −6.688489565971757425180382393062, −6.16187744937547398212159960916, −5.51307609210647112807653051082, −4.72367748362145126594809106254, −4.16095274683785618987064037996, −2.38312568500505151263623432286, −1.43207249684047130912436676636, 0.23398047274508976794357503275, 1.39141387608905999912100022590, 2.261992737766317250861028197436, 3.328888140997924197722111738705, 4.43092189812620583526332582184, 5.02380127664705489489365134574, 5.86813826448172677334994727678, 6.75327390754809495132896624831, 7.64831388710565354742577323193, 8.91610657507976748583824122479, 10.00495261388776145617764949925, 10.39099191133402225453464445213, 10.83866205314134663610506018386, 11.85304987157080588255731869, 12.461610888309011571228550298958, 13.35142221572237146479007225962, 14.03756119031988624397596921348, 14.66959726225026992960493620646, 15.75484675011089013768036013035, 16.92778632231681166468816293461, 17.44336885467009596987230862164, 18.01845949614514601517251046516, 18.61075200793979805514252927960, 19.63760497890354598796564623242, 20.40236179711169784162160270193

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