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Properties

Label 1-1441-1441.1080-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.881 - 0.471i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.644 + 0.764i)2-s + (−0.906 − 0.421i)3-s + (−0.168 + 0.985i)4-s + (0.485 + 0.873i)5-s + (−0.262 − 0.964i)6-s + (−0.779 + 0.626i)7-s + (−0.861 + 0.506i)8-s + (0.644 + 0.764i)9-s + (−0.354 + 0.935i)10-s + (0.568 − 0.822i)12-s + (0.262 − 0.964i)13-s + (−0.981 − 0.192i)14-s + (−0.0724 − 0.997i)15-s + (−0.943 − 0.331i)16-s + (−0.943 − 0.331i)17-s + (−0.168 + 0.985i)18-s + ⋯
L(s)  = 1  + (0.644 + 0.764i)2-s + (−0.906 − 0.421i)3-s + (−0.168 + 0.985i)4-s + (0.485 + 0.873i)5-s + (−0.262 − 0.964i)6-s + (−0.779 + 0.626i)7-s + (−0.861 + 0.506i)8-s + (0.644 + 0.764i)9-s + (−0.354 + 0.935i)10-s + (0.568 − 0.822i)12-s + (0.262 − 0.964i)13-s + (−0.981 − 0.192i)14-s + (−0.0724 − 0.997i)15-s + (−0.943 − 0.331i)16-s + (−0.943 − 0.331i)17-s + (−0.168 + 0.985i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1080, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6911881672 - 0.1732452820i\)
\(L(\frac12)\) \(\approx\) \(0.6911881672 - 0.1732452820i\)
\(L(1)\) \(\approx\) \(0.8171929423 + 0.3599850092i\)
\(L(1)\) \(\approx\) \(0.8171929423 + 0.3599850092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.644 + 0.764i)T \)
3 \( 1 + (-0.906 - 0.421i)T \)
5 \( 1 + (0.485 + 0.873i)T \)
7 \( 1 + (-0.779 + 0.626i)T \)
13 \( 1 + (0.262 - 0.964i)T \)
17 \( 1 + (-0.943 - 0.331i)T \)
19 \( 1 + (-0.607 - 0.794i)T \)
23 \( 1 + (0.748 - 0.663i)T \)
29 \( 1 + (-0.527 - 0.849i)T \)
31 \( 1 + (-0.215 - 0.976i)T \)
37 \( 1 + (0.168 - 0.985i)T \)
41 \( 1 + (0.607 + 0.794i)T \)
43 \( 1 + (-0.120 + 0.992i)T \)
47 \( 1 + (0.527 - 0.849i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.607 + 0.794i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.120 - 0.992i)T \)
71 \( 1 + (-0.995 - 0.0965i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.681 + 0.732i)T \)
83 \( 1 + (-0.443 + 0.896i)T \)
89 \( 1 + T \)
97 \( 1 + (0.262 - 0.964i)T \)
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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

−20.73106715035346769454849568191, −20.42033689534923433465807485609, −19.348284650106199480834426123383, −18.73788146115109794946534057963, −17.6111317893732525370186846029, −17.038848629168365106759301667880, −16.20154724304424149131418421341, −15.67880129215185222441594989913, −14.56871341471256605181792412094, −13.67381002247784792863201529278, −12.97227299068871401552831635299, −12.474437064227515999373896926235, −11.63969420581986020217220567395, −10.74328399737435550613961726089, −10.269263102272614984448612154050, −9.277938135655256734225189098010, −8.92164055660365398099315265528, −7.06077376946872135638613394852, −6.31574357774360977909782856559, −5.66636523389512372094495240804, −4.70312630501464702133017908269, −4.16037987427678051704766402811, −3.33862240275978897714044718186, −1.84359606857186092875232575159, −1.11965987804551530512589636814, 0.258215136201268989185075146915, 2.28853296809097392799398593519, 2.81305695407174838036256732557, 4.040397344208039452863870440918, 5.07707439592696270403163458074, 5.89848621193088385740226068086, 6.358402851810091899901811655320, 7.00182311979533881116780100046, 7.83713479300503143003133548332, 8.96220102832016698552504827785, 9.85005523003968365267770255145, 11.00640594825486058941221512743, 11.40190358948707803192091004564, 12.63432715022318226879284970676, 13.05253007076333768224634828868, 13.590560539028364423106089635113, 14.82096054151125814536460940484, 15.34782241461174993707406279139, 16.042517704835854608181447672530, 16.94212718820171827913021449598, 17.60153129468848481618051611445, 18.22146297781302124058226960751, 18.81581269808804886822761608799, 19.833840624307875740860008700190, 21.10726464280878252152637284093

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