↑

Properties

Label 1-1441-1441.522-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.700 + 0.713i$
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  + (0.836 − 0.548i)2-s + (0.0241 − 0.999i)3-s + (0.399 − 0.916i)4-s + (0.958 − 0.285i)5-s + (−0.527 − 0.849i)6-s + (−0.970 − 0.239i)7-s + (−0.168 − 0.985i)8-s + (−0.998 − 0.0483i)9-s + (0.644 − 0.764i)10-s + (−0.906 − 0.421i)12-s + (−0.970 − 0.239i)13-s + (−0.943 + 0.331i)14-s + (−0.262 − 0.964i)15-s + (−0.681 − 0.732i)16-s + (−0.906 + 0.421i)17-s + (−0.861 + 0.506i)18-s + ⋯
L(s)  = 1  + (0.836 − 0.548i)2-s + (0.0241 − 0.999i)3-s + (0.399 − 0.916i)4-s + (0.958 − 0.285i)5-s + (−0.527 − 0.849i)6-s + (−0.970 − 0.239i)7-s + (−0.168 − 0.985i)8-s + (−0.998 − 0.0483i)9-s + (0.644 − 0.764i)10-s + (−0.906 − 0.421i)12-s + (−0.970 − 0.239i)13-s + (−0.943 + 0.331i)14-s + (−0.262 − 0.964i)15-s + (−0.681 − 0.732i)16-s + (−0.906 + 0.421i)17-s + (−0.861 + 0.506i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.700 + 0.713i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (522, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.700 + 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6836888514 - 1.628048854i\)
\(L(\frac12)\) \(\approx\) \(-0.6836888514 - 1.628048854i\)
\(L(1)\) \(\approx\) \(0.8526483587 - 1.191828742i\)
\(L(1)\) \(\approx\) \(0.8526483587 - 1.191828742i\)

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 \( 1 + (0.0241 - 0.999i)T \)
5 \( 1 + (0.958 - 0.285i)T \)
7 \( 1 + (-0.970 - 0.239i)T \)
13 \( 1 + (-0.970 - 0.239i)T \)
17 \( 1 + (-0.906 + 0.421i)T \)
19 \( 1 + (0.981 - 0.192i)T \)
23 \( 1 + (-0.443 - 0.896i)T \)
29 \( 1 + (-0.354 - 0.935i)T \)
31 \( 1 + (0.715 + 0.698i)T \)
37 \( 1 + (-0.748 - 0.663i)T \)
41 \( 1 + (0.485 + 0.873i)T \)
43 \( 1 + (0.958 - 0.285i)T \)
47 \( 1 + (-0.998 - 0.0483i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.906 + 0.421i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.958 + 0.285i)T \)
71 \( 1 + (-0.989 - 0.144i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.943 - 0.331i)T \)
83 \( 1 + (0.399 + 0.916i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.0724 - 0.997i)T \)
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

−21.54501827208551600081732198041, −20.5985576861173505855344147432, −20.06477117841377727928238716177, −19.02188810583638641027234615151, −17.78217114800633626361510200490, −17.27023544705855174020752988033, −16.463768621565016487505324819856, −15.79970012789925263980498224547, −15.25021204676545673866681128792, −14.26483882655170584226024395195, −13.8679052273573899245847847452, −13.01551377881438784539164480157, −12.11103028296647069034079811975, −11.30545477726435839653876766040, −10.34100027522966670644506894252, −9.434718557726390455473599074149, −9.14174767503546239913218627202, −7.7730016608289049192922363939, −6.81591834511615750055040895833, −6.09377948871629250554474224473, −5.338071669424603164451155120037, −4.69974259189567853129493174567, −3.55893339070041566400008490596, −2.91776554129254057670570781356, −2.11722531072919104674118848577, 0.43838275079069941856308948108, 1.55156541813139032978364717240, 2.4787362967135551083326314926, 2.970722310200099140808299619903, 4.28313224188893288858850088117, 5.271743884457541850481855989587, 6.06639259795217494837629258670, 6.61932520322854362436381550655, 7.409589650811769095488035206666, 8.725481202266091410459594152157, 9.63155079130931713175056329038, 10.21102553304838293419429946976, 11.20643223711691854516998618064, 12.189528835156092037118703589486, 12.72508243083481714102780673436, 13.25839783972174731517334253694, 13.95458103552279933278029044633, 14.5129920954228122685203185153, 15.63104244537501400321147183731, 16.48957591510583819487674963003, 17.428274092897611565284971875097, 18.03120158431057808033068975898, 19.00723691046232577986250061981, 19.64098757354912255123327227882, 20.15445368130487678988024380652

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