↑

Properties

Label 1-1441-1441.118-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.999 + 0.0313i$
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  + (0.926 − 0.377i)2-s + (−0.681 + 0.732i)3-s + (0.715 − 0.698i)4-s + (−0.906 − 0.421i)5-s + (−0.354 + 0.935i)6-s + (−0.779 + 0.626i)7-s + (0.399 − 0.916i)8-s + (−0.0724 − 0.997i)9-s + (−0.998 − 0.0483i)10-s + (0.0241 + 0.999i)12-s + (0.262 − 0.964i)13-s + (−0.485 + 0.873i)14-s + (0.926 − 0.377i)15-s + (0.0241 − 0.999i)16-s + (0.568 + 0.822i)17-s + (−0.443 − 0.896i)18-s + ⋯
L(s)  = 1  + (0.926 − 0.377i)2-s + (−0.681 + 0.732i)3-s + (0.715 − 0.698i)4-s + (−0.906 − 0.421i)5-s + (−0.354 + 0.935i)6-s + (−0.779 + 0.626i)7-s + (0.399 − 0.916i)8-s + (−0.0724 − 0.997i)9-s + (−0.998 − 0.0483i)10-s + (0.0241 + 0.999i)12-s + (0.262 − 0.964i)13-s + (−0.485 + 0.873i)14-s + (0.926 − 0.377i)15-s + (0.0241 − 0.999i)16-s + (0.568 + 0.822i)17-s + (−0.443 − 0.896i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.999 + 0.0313i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.999 + 0.0313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004297243754 - 0.2744035174i\)
\(L(\frac12)\) \(\approx\) \(0.004297243754 - 0.2744035174i\)
\(L(1)\) \(\approx\) \(0.9314904549 - 0.1587259022i\)
\(L(1)\) \(\approx\) \(0.9314904549 - 0.1587259022i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.926 - 0.377i)T \)
3 \( 1 + (-0.681 + 0.732i)T \)
5 \( 1 + (-0.906 - 0.421i)T \)
7 \( 1 + (-0.779 + 0.626i)T \)
13 \( 1 + (0.262 - 0.964i)T \)
17 \( 1 + (0.568 + 0.822i)T \)
19 \( 1 + (0.0241 + 0.999i)T \)
23 \( 1 + (-0.399 - 0.916i)T \)
29 \( 1 + (-0.527 - 0.849i)T \)
31 \( 1 + (-0.399 + 0.916i)T \)
37 \( 1 + (0.168 - 0.985i)T \)
41 \( 1 + (-0.568 + 0.822i)T \)
43 \( 1 + (-0.485 - 0.873i)T \)
47 \( 1 + (-0.926 + 0.377i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.958 - 0.285i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.485 + 0.873i)T \)
71 \( 1 + (-0.399 + 0.916i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.906 - 0.421i)T \)
83 \( 1 + (-0.168 - 0.985i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.779 + 0.626i)T \)
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

−21.36260025176120736130745095266, −20.2062903230363478487923940091, −19.677066894492401905166892621197, −18.885517666542410942399908075999, −18.12488158650697479713215555866, −17.05351378291762264464993182370, −16.42549532299491813938992286948, −15.97100436136197632366719970232, −15.05970753270802067030226123605, −14.05987997835702078293530620812, −13.49456299849373048269270915564, −12.81075544961884383876620679684, −11.8796610999860765579955241632, −11.45354229658786445515059113357, −10.79360218306303258723695721998, −9.57445937715879240180499687162, −8.23534201378426125374861001141, −7.385055794418917375189974830569, −6.9455958094992275337877994539, −6.35719274554894853520920786169, −5.294486417305763159839559860244, −4.44540534612124301438464351915, −3.55475354984950825475382209050, −2.77638008392354229976423737878, −1.48761193739118427523382571115, 0.082762144568497156193174522877, 1.421588037294050003624456972581, 2.95932523308937627132582405142, 3.61812664768769362624553857125, 4.24176011285418877408005805742, 5.29634505376187495613164391967, 5.82974952685405991469002067362, 6.57977284555504878961252496669, 7.81072661531943094922468806672, 8.778765081070205379466894544478, 9.88729906957274209372179809362, 10.42468507087598785912745819940, 11.25447831462500906320248030497, 12.14359088385973838547249431475, 12.46703091570750470156804879284, 13.14175816005092660353122546868, 14.60839277661809435004321996782, 15.002127169053572914113879142616, 15.88176024128180398949717216366, 16.20407219043099921766059761938, 16.9903484838428422688492104287, 18.28897761403389623219408918032, 19.00636443318845066522470841565, 19.831727645679612814170618368363, 20.512624646790682227512838585485

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