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Properties

Label 1-1441-1441.670-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.401 + 0.916i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (−0.836 + 0.548i)2-s + (−0.607 + 0.794i)3-s + (0.399 − 0.916i)4-s + (0.0241 − 0.999i)5-s + (0.0724 − 0.997i)6-s + (−0.644 − 0.764i)7-s + (0.168 + 0.985i)8-s + (−0.262 − 0.964i)9-s + (0.527 + 0.849i)10-s + (0.485 + 0.873i)12-s + (0.527 − 0.849i)13-s + (0.958 + 0.285i)14-s + (0.779 + 0.626i)15-s + (−0.681 − 0.732i)16-s + (−0.981 − 0.192i)17-s + (0.748 + 0.663i)18-s + ⋯
L(s)  = 1  + (−0.836 + 0.548i)2-s + (−0.607 + 0.794i)3-s + (0.399 − 0.916i)4-s + (0.0241 − 0.999i)5-s + (0.0724 − 0.997i)6-s + (−0.644 − 0.764i)7-s + (0.168 + 0.985i)8-s + (−0.262 − 0.964i)9-s + (0.527 + 0.849i)10-s + (0.485 + 0.873i)12-s + (0.527 − 0.849i)13-s + (0.958 + 0.285i)14-s + (0.779 + 0.626i)15-s + (−0.681 − 0.732i)16-s + (−0.981 − 0.192i)17-s + (0.748 + 0.663i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.401 + 0.916i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (670, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.401 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4067693198 + 0.2659565220i\)
\(L(\frac12)\) \(\approx\) \(0.4067693198 + 0.2659565220i\)
\(L(1)\) \(\approx\) \(0.5006541767 + 0.05629350963i\)
\(L(1)\) \(\approx\) \(0.5006541767 + 0.05629350963i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.836 + 0.548i)T \)
3 \( 1 + (-0.607 + 0.794i)T \)
5 \( 1 + (0.0241 - 0.999i)T \)
7 \( 1 + (-0.644 - 0.764i)T \)
13 \( 1 + (0.527 - 0.849i)T \)
17 \( 1 + (-0.981 - 0.192i)T \)
19 \( 1 + (-0.120 + 0.992i)T \)
23 \( 1 + (0.715 - 0.698i)T \)
29 \( 1 + (0.262 - 0.964i)T \)
31 \( 1 + (-0.443 + 0.896i)T \)
37 \( 1 + (0.215 + 0.976i)T \)
41 \( 1 + (0.681 - 0.732i)T \)
43 \( 1 + (0.943 + 0.331i)T \)
47 \( 1 + (-0.354 + 0.935i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.906 + 0.421i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.943 + 0.331i)T \)
71 \( 1 + (0.885 - 0.464i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.568 - 0.822i)T \)
83 \( 1 + (-0.995 + 0.0965i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.926 - 0.377i)T \)
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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

−19.910899259105206803193624329922, −19.50360350342730894622655999344, −18.68950349426591324484699841650, −18.37745729312052138540234337211, −17.62907844428962955642148127623, −16.933616638374843953578430924033, −15.94214798648923101390358091748, −15.42754274269834955406975431232, −14.15447035647681099581896361978, −13.180391796055330320035766492983, −12.727038464789129943335430385, −11.65816344563358629484662187243, −11.18016121827341159436105638370, −10.684840272145007388404696319292, −9.43188799667285249290505605631, −8.9494319421864072522301913, −7.84454720368656777952099086767, −6.90532254020688482432066411625, −6.61233435459508978137126271998, −5.68105772742321485747435255817, −4.23790053300357653649130955481, −3.06012314968644873145170207545, −2.38013338121106743212002668393, −1.59326992256925363547221048642, −0.24186080375587484061147510295, 0.56292314109295555931510639495, 1.33903063873042025821351964306, 2.96359139722732801650014171627, 4.19512257218191671170548899422, 4.828509598392445856577493701060, 5.884336437117750000102996996011, 6.32907134794528884146179686025, 7.44555336927196714455729912333, 8.39502890189798159672358011208, 9.09378594463056051844848539002, 9.79999802854669936609674329347, 10.53479775738156928188147599638, 11.07869916959671689650271597583, 12.19512293703885094502384623886, 13.01263175271997250923913924896, 13.95154657964685929981140795355, 14.97527891218335077012924745273, 15.82758545278601474436215521464, 16.15435709508487406369448587323, 16.89303601855764526209909773041, 17.43432596964319561292349730649, 18.09920004300872299961404065431, 19.23154686700360255546356441125, 19.9660402529745427511372119106, 20.66168583395202266213192093216

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