↑

Properties

Label 1-1441-1441.128-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.125 - 0.992i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0241 − 0.999i)2-s + (−0.989 + 0.144i)3-s + (−0.998 − 0.0483i)4-s + (−0.168 + 0.985i)5-s + (0.120 + 0.992i)6-s + (0.906 + 0.421i)7-s + (−0.0724 + 0.997i)8-s + (0.958 − 0.285i)9-s + (0.981 + 0.192i)10-s + (0.995 − 0.0965i)12-s + (−0.485 − 0.873i)13-s + (0.443 − 0.896i)14-s + (0.0241 − 0.999i)15-s + (0.995 + 0.0965i)16-s + (−0.748 − 0.663i)17-s + (−0.262 − 0.964i)18-s + ⋯
L(s)  = 1  + (0.0241 − 0.999i)2-s + (−0.989 + 0.144i)3-s + (−0.998 − 0.0483i)4-s + (−0.168 + 0.985i)5-s + (0.120 + 0.992i)6-s + (0.906 + 0.421i)7-s + (−0.0724 + 0.997i)8-s + (0.958 − 0.285i)9-s + (0.981 + 0.192i)10-s + (0.995 − 0.0965i)12-s + (−0.485 − 0.873i)13-s + (0.443 − 0.896i)14-s + (0.0241 − 0.999i)15-s + (0.995 + 0.0965i)16-s + (−0.748 − 0.663i)17-s + (−0.262 − 0.964i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.125 - 0.992i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.125 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5331682083 - 0.6048308741i\)
\(L(\frac12)\) \(\approx\) \(0.5331682083 - 0.6048308741i\)
\(L(1)\) \(\approx\) \(0.6784116411 - 0.2592337036i\)
\(L(1)\) \(\approx\) \(0.6784116411 - 0.2592337036i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.0241 - 0.999i)T \)
3 \( 1 + (-0.989 + 0.144i)T \)
5 \( 1 + (-0.168 + 0.985i)T \)
7 \( 1 + (0.906 + 0.421i)T \)
13 \( 1 + (-0.485 - 0.873i)T \)
17 \( 1 + (-0.748 - 0.663i)T \)
19 \( 1 + (0.995 - 0.0965i)T \)
23 \( 1 + (0.0724 + 0.997i)T \)
29 \( 1 + (-0.607 - 0.794i)T \)
31 \( 1 + (0.0724 - 0.997i)T \)
37 \( 1 + (-0.779 - 0.626i)T \)
41 \( 1 + (0.748 - 0.663i)T \)
43 \( 1 + (0.443 + 0.896i)T \)
47 \( 1 + (-0.0241 + 0.999i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.399 - 0.916i)T \)
61 \( 1 - T \)
67 \( 1 + (0.443 - 0.896i)T \)
71 \( 1 + (0.0724 - 0.997i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.168 + 0.985i)T \)
83 \( 1 + (0.779 - 0.626i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.906 + 0.421i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.13967898142234619742653155406, −20.20996033347500931077442784995, −19.227863970399289051069571015269, −18.31438947598757261427285304635, −17.67950886823819694210507181039, −17.02274719387016548132053030152, −16.5446746783215337040360226139, −15.89675834567076735031585379744, −15.0140248408345482560531087159, −14.09361552234236571584685875875, −13.3984757334193154066503607211, −12.4953007307247765754584228110, −11.96206157247112743423015264431, −10.96765249676607462214224535323, −10.08772320113111233634877916082, −9.06502099581673068576839833994, −8.40811620573163983745782870538, −7.44164633830756564827030296135, −6.89495449314363624207297584514, −5.87984859333735522800995541947, −5.005174830697801836047166231489, −4.637290147479373842814552436982, −3.84571932406512075867966518211, −1.75489241212518332534390315487, −0.9292269285815899791515725814, 0.45730770315019655369918821196, 1.75743355026550099336540875191, 2.640232117033460195216442452932, 3.62182527328843899269547218832, 4.586757119681649191575951172530, 5.34642782081048348234235778801, 6.02371101654755212306254930480, 7.39833194043181568478093291757, 7.87118034390124309676644244932, 9.36228548218910348640112253079, 9.808809389713168825165493866503, 10.978100146393924436024809421718, 11.13783236487182163566834497574, 11.83264407177772059833499582227, 12.60409201486685816453871804343, 13.57665777900517146158598100192, 14.359873329168918746946512536101, 15.26337686396559632834076661864, 15.79528732065031002629679861010, 17.31025883415536313192169962743, 17.648646296215720815683942089379, 18.23363820240498667338534041165, 18.89582190987175446725426476336, 19.747110561535264090369654351067, 20.71695533655252956144309800282

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