Properties

Label 1-1441-1441.434-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: $\chi_{1441} (434, \cdot )$
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

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

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