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Properties

Label 1-1441-1441.1018-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.177 - 0.984i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (−0.399 + 0.916i)2-s + (0.779 + 0.626i)3-s + (−0.681 − 0.732i)4-s + (−0.262 − 0.964i)5-s + (−0.885 + 0.464i)6-s + (0.443 + 0.896i)7-s + (0.943 − 0.331i)8-s + (0.215 + 0.976i)9-s + (0.989 + 0.144i)10-s + (−0.0724 − 0.997i)12-s + (−0.715 + 0.698i)13-s + (−0.998 + 0.0483i)14-s + (0.399 − 0.916i)15-s + (−0.0724 + 0.997i)16-s + (0.970 − 0.239i)17-s + (−0.981 − 0.192i)18-s + ⋯
L(s)  = 1  + (−0.399 + 0.916i)2-s + (0.779 + 0.626i)3-s + (−0.681 − 0.732i)4-s + (−0.262 − 0.964i)5-s + (−0.885 + 0.464i)6-s + (0.443 + 0.896i)7-s + (0.943 − 0.331i)8-s + (0.215 + 0.976i)9-s + (0.989 + 0.144i)10-s + (−0.0724 − 0.997i)12-s + (−0.715 + 0.698i)13-s + (−0.998 + 0.0483i)14-s + (0.399 − 0.916i)15-s + (−0.0724 + 0.997i)16-s + (0.970 − 0.239i)17-s + (−0.981 − 0.192i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.177 - 0.984i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1018, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.177 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3182649819 + 0.3807832188i\)
\(L(\frac12)\) \(\approx\) \(-0.3182649819 + 0.3807832188i\)
\(L(1)\) \(\approx\) \(0.7160913930 + 0.5573997776i\)
\(L(1)\) \(\approx\) \(0.7160913930 + 0.5573997776i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.399 + 0.916i)T \)
3 \( 1 + (0.779 + 0.626i)T \)
5 \( 1 + (-0.262 - 0.964i)T \)
7 \( 1 + (0.443 + 0.896i)T \)
13 \( 1 + (-0.715 + 0.698i)T \)
17 \( 1 + (0.970 - 0.239i)T \)
19 \( 1 + (0.0724 + 0.997i)T \)
23 \( 1 + (-0.943 - 0.331i)T \)
29 \( 1 + (-0.995 + 0.0965i)T \)
31 \( 1 + (-0.943 + 0.331i)T \)
37 \( 1 + (0.485 - 0.873i)T \)
41 \( 1 + (0.970 + 0.239i)T \)
43 \( 1 + (0.998 + 0.0483i)T \)
47 \( 1 + (0.399 - 0.916i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.644 - 0.764i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.998 + 0.0483i)T \)
71 \( 1 + (-0.943 + 0.331i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.262 + 0.964i)T \)
83 \( 1 + (-0.485 - 0.873i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.443 - 0.896i)T \)
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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

−19.870505247976600582053703748054, −19.37236368497427596214141724332, −18.65720691777600306712152454200, −17.84057883690686284072196348544, −17.49794906540024575311334212783, −16.40373744402540741123057618847, −15.10967227027535731256075222545, −14.51063599467118458109109117704, −13.822170679860322717174951966693, −13.12144711833872085188763649657, −12.29498675229306808911818911468, −11.48298425839068553860375165849, −10.736612324223301782845594401302, −9.97321233494308566429254789265, −9.30233466811479097945371028308, −8.0986662275682016584596597414, −7.562531244887372549763267763582, −7.19301979904898369881440872699, −5.7900399367731138087561586464, −4.36165149302917059888743311595, −3.63226252643888582214791719321, −2.87905859289937821126125296281, −2.10850946628756377217281440052, −1.069610999151746509351670144498, −0.10133828563582285236845037809, 1.4327381517193798704240240034, 2.253021641253960104436328829697, 3.79942444454239622653869722156, 4.45179203988919342634585319763, 5.370165475719173899071357174, 5.82406840532894406491248359610, 7.48992186031074195114939555826, 7.82049303681132937883193701841, 8.71971784746172562879679962583, 9.2762757563212329543847430656, 9.80284669518298725706774064106, 10.88160318677581627309954215313, 12.06628277584776704789024770761, 12.72742548428206555581498001138, 13.883751071087996915411489944673, 14.511719712468222656478460766048, 14.97786326961754185119382056808, 15.99192568266741771909136734527, 16.37254637346473761777714035792, 16.981670021887199045058362614990, 18.12889993849448123636737033041, 18.81251003376188766155148322615, 19.49907529978300108517259552280, 20.26831300081862522100445314717, 21.061786711359402985052060189172

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