Properties

Label 1-1441-1441.933-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.304 - 0.952i$
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  + (−0.681 − 0.732i)2-s + (0.995 + 0.0965i)3-s + (−0.0724 + 0.997i)4-s + (0.399 − 0.916i)5-s + (−0.607 − 0.794i)6-s + (0.568 + 0.822i)7-s + (0.779 − 0.626i)8-s + (0.981 + 0.192i)9-s + (−0.943 + 0.331i)10-s + (−0.168 + 0.985i)12-s + (0.568 + 0.822i)13-s + (0.215 − 0.976i)14-s + (0.485 − 0.873i)15-s + (−0.989 − 0.144i)16-s + (−0.168 − 0.985i)17-s + (−0.527 − 0.849i)18-s + ⋯
L(s)  = 1  + (−0.681 − 0.732i)2-s + (0.995 + 0.0965i)3-s + (−0.0724 + 0.997i)4-s + (0.399 − 0.916i)5-s + (−0.607 − 0.794i)6-s + (0.568 + 0.822i)7-s + (0.779 − 0.626i)8-s + (0.981 + 0.192i)9-s + (−0.943 + 0.331i)10-s + (−0.168 + 0.985i)12-s + (0.568 + 0.822i)13-s + (0.215 − 0.976i)14-s + (0.485 − 0.873i)15-s + (−0.989 − 0.144i)16-s + (−0.168 − 0.985i)17-s + (−0.527 − 0.849i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.304 - 0.952i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (933, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.304 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.582690937 - 1.156061665i\)
\(L(\frac12)\) \(\approx\) \(1.582690937 - 1.156061665i\)
\(L(1)\) \(\approx\) \(1.189226211 - 0.4797753185i\)
\(L(1)\) \(\approx\) \(1.189226211 - 0.4797753185i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.681 - 0.732i)T \)
3 \( 1 + (0.995 + 0.0965i)T \)
5 \( 1 + (0.399 - 0.916i)T \)
7 \( 1 + (0.568 + 0.822i)T \)
13 \( 1 + (0.568 + 0.822i)T \)
17 \( 1 + (-0.168 - 0.985i)T \)
19 \( 1 + (0.715 - 0.698i)T \)
23 \( 1 + (-0.262 - 0.964i)T \)
29 \( 1 + (0.120 - 0.992i)T \)
31 \( 1 + (-0.998 + 0.0483i)T \)
37 \( 1 + (-0.970 + 0.239i)T \)
41 \( 1 + (-0.443 - 0.896i)T \)
43 \( 1 + (0.399 - 0.916i)T \)
47 \( 1 + (0.981 + 0.192i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.168 - 0.985i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.399 + 0.916i)T \)
71 \( 1 + (0.836 + 0.548i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.215 + 0.976i)T \)
83 \( 1 + (-0.0724 - 0.997i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.958 - 0.285i)T \)
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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

−20.64963239489876637656889049772, −19.91862475929876186809947298303, −19.39129241152525519833926791834, −18.338028218434801996245014437179, −18.0615418506775880775214593580, −17.288902181243706384856777802143, −16.3011823824407193871460270906, −15.42737406806736768609635623012, −14.833444591061385315796582878473, −14.17194056412599674551670182965, −13.69219297673713435560671262321, −12.82653010224390009253072994844, −11.27660037203099701030155274510, −10.51861110820029430021425590823, −10.073879629284876961011540997312, −9.16469472287069756566078435938, −8.237510589725643469515578839068, −7.64282793929017616064255153231, −7.06792599367732907094419115483, −6.12676043765180613658556405740, −5.242524236891443874865488352501, −3.92406446789573269179436108808, −3.21626813872751380992926212892, −1.84121460173217551564240544748, −1.31040859815870194418138518709, 0.89647596692301107050658774042, 2.00477308708381222393601483785, 2.3546485441795988851205124948, 3.56124801643088401647157054969, 4.479414171373257669109967208358, 5.21508937135719079137677632041, 6.67450444360869688517887131990, 7.67352438858351312020655626715, 8.45076570823282518988528175469, 9.135976650842332652255942418767, 9.306530831328655002597228020316, 10.39954183227461705717536559432, 11.41367333824665419032828930606, 12.13727912463769999762584035875, 12.828422396527330435679223862770, 13.812104407599998891396538811635, 14.10379558402388604708250401151, 15.692734330509696257037240356230, 15.86203383958660935799214389326, 16.95218108519573254646932393486, 17.74918393106598465615165806170, 18.61665533390601674561837591457, 18.91056003291323427356364706219, 20.07747577337937110254226196413, 20.56146475455370359530430089226

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