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Properties

Label 1-1441-1441.567-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.844 - 0.535i$
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  + (0.607 − 0.794i)2-s + (0.715 − 0.698i)3-s + (−0.262 − 0.964i)4-s + (−0.989 + 0.144i)5-s + (−0.120 − 0.992i)6-s + (0.681 − 0.732i)7-s + (−0.926 − 0.377i)8-s + (0.0241 − 0.999i)9-s + (−0.485 + 0.873i)10-s + (−0.861 − 0.506i)12-s + (0.906 + 0.421i)13-s + (−0.168 − 0.985i)14-s + (−0.607 + 0.794i)15-s + (−0.861 + 0.506i)16-s + (0.748 + 0.663i)17-s + (−0.779 − 0.626i)18-s + ⋯
L(s)  = 1  + (0.607 − 0.794i)2-s + (0.715 − 0.698i)3-s + (−0.262 − 0.964i)4-s + (−0.989 + 0.144i)5-s + (−0.120 − 0.992i)6-s + (0.681 − 0.732i)7-s + (−0.926 − 0.377i)8-s + (0.0241 − 0.999i)9-s + (−0.485 + 0.873i)10-s + (−0.861 − 0.506i)12-s + (0.906 + 0.421i)13-s + (−0.168 − 0.985i)14-s + (−0.607 + 0.794i)15-s + (−0.861 + 0.506i)16-s + (0.748 + 0.663i)17-s + (−0.779 − 0.626i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.844 - 0.535i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.844 - 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.244625165 - 4.287211133i\)
\(L(\frac12)\) \(\approx\) \(1.244625165 - 4.287211133i\)
\(L(1)\) \(\approx\) \(1.301093314 - 1.402658274i\)
\(L(1)\) \(\approx\) \(1.301093314 - 1.402658274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.607 - 0.794i)T \)
3 \( 1 + (0.715 - 0.698i)T \)
5 \( 1 + (-0.989 + 0.144i)T \)
7 \( 1 + (0.681 - 0.732i)T \)
13 \( 1 + (0.906 + 0.421i)T \)
17 \( 1 + (0.748 + 0.663i)T \)
19 \( 1 + (0.861 + 0.506i)T \)
23 \( 1 + (0.926 - 0.377i)T \)
29 \( 1 + (0.943 - 0.331i)T \)
31 \( 1 + (0.926 + 0.377i)T \)
37 \( 1 + (0.836 - 0.548i)T \)
41 \( 1 + (0.748 - 0.663i)T \)
43 \( 1 + (0.168 - 0.985i)T \)
47 \( 1 + (-0.607 + 0.794i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.995 + 0.0965i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.168 - 0.985i)T \)
71 \( 1 + (0.926 + 0.377i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.989 - 0.144i)T \)
83 \( 1 + (-0.836 - 0.548i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.681 + 0.732i)T \)
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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

−21.04179164790465279057388420906, −20.32949948498447186620525205030, −19.46035130460512300256094837051, −18.483673041729598503487558777, −17.85355845126890713772301627804, −16.62901073853866035940341910580, −16.072857007611084651143510469405, −15.435613606549004056147093805704, −14.98704747776491934370463863008, −14.22412536361809296653964653664, −13.45051258508247856210248092019, −12.60891712211313991921145070053, −11.53520604677464550434952600208, −11.251583858578992579580843036039, −9.766395147421204500758711891811, −8.9198689544303122165808494648, −8.18165857369587195242954997810, −7.84644602421175247093793543946, −6.81757147874733628088651557186, −5.52649884597469321069838706837, −4.92756252026761596542357103375, −4.237140452938743587695779775900, −3.16498655264672735437000244016, −2.8156783566623122778133330773, −0.998929622768325714090611371940, 0.872209259598893686717856864571, 1.16848976107531336417409319725, 2.43732402929008135665207725664, 3.425411491499740046302103546196, 3.917893048416515075844609891144, 4.77934998578616159504369220675, 6.0437180794174292614326800172, 6.90823727665695543649902952260, 7.80980881647556099683085023114, 8.43894330384306046843591607706, 9.377022485759303616174442119511, 10.505821060258908473106995854065, 11.09205487895018409646881956156, 12.00836714348531735432574741846, 12.42382025138964402583724836100, 13.48454808180026934316201640138, 14.0350302115891162943241884755, 14.63996163288302677645043742338, 15.366665262844259702460880640006, 16.27537494985272813459252068942, 17.497824752101947823335662248090, 18.37255412848165832974855982165, 18.93551536976547013372862649857, 19.56954459822997371646866683334, 20.20704350629792557576672027452

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