Properties

Label 1-1441-1441.15-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.237 - 0.971i$
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  + (0.779 + 0.626i)2-s + (0.958 − 0.285i)3-s + (0.215 + 0.976i)4-s + (−0.943 − 0.331i)5-s + (0.926 + 0.377i)6-s + (−0.970 − 0.239i)7-s + (−0.443 + 0.896i)8-s + (0.836 − 0.548i)9-s + (−0.527 − 0.849i)10-s + (0.485 + 0.873i)12-s + (−0.970 − 0.239i)13-s + (−0.607 − 0.794i)14-s + (−0.998 − 0.0483i)15-s + (−0.906 + 0.421i)16-s + (0.485 − 0.873i)17-s + (0.995 + 0.0965i)18-s + ⋯
L(s)  = 1  + (0.779 + 0.626i)2-s + (0.958 − 0.285i)3-s + (0.215 + 0.976i)4-s + (−0.943 − 0.331i)5-s + (0.926 + 0.377i)6-s + (−0.970 − 0.239i)7-s + (−0.443 + 0.896i)8-s + (0.836 − 0.548i)9-s + (−0.527 − 0.849i)10-s + (0.485 + 0.873i)12-s + (−0.970 − 0.239i)13-s + (−0.607 − 0.794i)14-s + (−0.998 − 0.0483i)15-s + (−0.906 + 0.421i)16-s + (0.485 − 0.873i)17-s + (0.995 + 0.0965i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.237 - 0.971i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.237 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.138815408 - 0.8942687164i\)
\(L(\frac12)\) \(\approx\) \(1.138815408 - 0.8942687164i\)
\(L(1)\) \(\approx\) \(1.393256694 + 0.09740822496i\)
\(L(1)\) \(\approx\) \(1.393256694 + 0.09740822496i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.779 + 0.626i)T \)
3 \( 1 + (0.958 - 0.285i)T \)
5 \( 1 + (-0.943 - 0.331i)T \)
7 \( 1 + (-0.970 - 0.239i)T \)
13 \( 1 + (-0.970 - 0.239i)T \)
17 \( 1 + (0.485 - 0.873i)T \)
19 \( 1 + (-0.681 + 0.732i)T \)
23 \( 1 + (0.715 - 0.698i)T \)
29 \( 1 + (-0.354 - 0.935i)T \)
31 \( 1 + (-0.989 - 0.144i)T \)
37 \( 1 + (-0.748 - 0.663i)T \)
41 \( 1 + (0.981 - 0.192i)T \)
43 \( 1 + (-0.943 - 0.331i)T \)
47 \( 1 + (0.836 - 0.548i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.485 - 0.873i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.943 + 0.331i)T \)
71 \( 1 + (-0.168 - 0.985i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.607 + 0.794i)T \)
83 \( 1 + (0.215 - 0.976i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.644 + 0.764i)T \)
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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

−20.895637524804340973339416292874, −19.86630918777629973461888087498, −19.60392852865223150706476943235, −19.08983364809360470524372900861, −18.40298321965242145949622750326, −16.8439480142730414239754428167, −16.02033908571523964914534448002, −15.20114928949654347817120211963, −14.925824024009821954516117729916, −14.1355847585424182923809563884, −13.08165405219653018616404531212, −12.66952962182663315816798348749, −11.85855221426416664242779422496, −10.81408910614188341213861635082, −10.24991924496709325873449317856, −9.31728999735340193912140108849, −8.70838680058056375876981499740, −7.35041575402595874930667087494, −6.94699085273548445024780393349, −5.713722331490657398405480722015, −4.63152218200710071265294443936, −3.939688254424558619466837373123, −3.12623108068118990672834329080, −2.68066734524247941439431789221, −1.47050342216980199777969011499, 0.34431670608052031833070833384, 2.17444030653000689324097650919, 3.081709575812070967539411020704, 3.72231347508930383028635398464, 4.45249601355181568135804779523, 5.46946328382838558092777290838, 6.649933221557247586817750243651, 7.31271300459731063381016031177, 7.79734165504186875665278288271, 8.739137439860061883874711803317, 9.43638877688434086735664269089, 10.53254184555549140240512409996, 11.833515769853759382648747495410, 12.47370481818565114290419425552, 12.92260417417702366840976880073, 13.74633517868697253691599340701, 14.6812342919141396362843054657, 15.047512739915315088301879087011, 15.97536468691092862240372183073, 16.48065940617136306878616580979, 17.271196371072423562576224054235, 18.562558277889529179536681768702, 19.16747445665224212247266135493, 19.93098135830762308174728730492, 20.55434554070563170738100152443

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