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Properties

Label 1-1441-1441.403-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

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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: $\chi_{1441} (403, \cdot )$
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 \( 1 + (-0.779 - 0.626i)T \)
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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

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

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