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Properties

Label 1-1441-1441.86-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.993 - 0.113i$
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  + (−0.926 − 0.377i)2-s + (0.981 + 0.192i)3-s + (0.715 + 0.698i)4-s + (−0.681 − 0.732i)5-s + (−0.836 − 0.548i)6-s + (−0.998 − 0.0483i)7-s + (−0.399 − 0.916i)8-s + (0.926 + 0.377i)9-s + (0.354 + 0.935i)10-s + (0.568 + 0.822i)12-s + (0.836 − 0.548i)13-s + (0.906 + 0.421i)14-s + (−0.527 − 0.849i)15-s + (0.0241 + 0.999i)16-s + (−0.0241 − 0.999i)17-s + (−0.715 − 0.698i)18-s + ⋯
L(s)  = 1  + (−0.926 − 0.377i)2-s + (0.981 + 0.192i)3-s + (0.715 + 0.698i)4-s + (−0.681 − 0.732i)5-s + (−0.836 − 0.548i)6-s + (−0.998 − 0.0483i)7-s + (−0.399 − 0.916i)8-s + (0.926 + 0.377i)9-s + (0.354 + 0.935i)10-s + (0.568 + 0.822i)12-s + (0.836 − 0.548i)13-s + (0.906 + 0.421i)14-s + (−0.527 − 0.849i)15-s + (0.0241 + 0.999i)16-s + (−0.0241 − 0.999i)17-s + (−0.715 − 0.698i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.993 - 0.113i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.993 - 0.113i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329644673 - 0.07544957214i\)
\(L(\frac12)\) \(\approx\) \(1.329644673 - 0.07544957214i\)
\(L(1)\) \(\approx\) \(0.8010617979 - 0.1420808526i\)
\(L(1)\) \(\approx\) \(0.8010617979 - 0.1420808526i\)

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 \( 1 + (0.981 + 0.192i)T \)
5 \( 1 + (-0.681 - 0.732i)T \)
7 \( 1 + (-0.998 - 0.0483i)T \)
13 \( 1 + (0.836 - 0.548i)T \)
17 \( 1 + (-0.0241 - 0.999i)T \)
19 \( 1 + (-0.958 + 0.285i)T \)
23 \( 1 + (0.748 + 0.663i)T \)
29 \( 1 + (0.0724 + 0.997i)T \)
31 \( 1 + (-0.995 + 0.0965i)T \)
37 \( 1 + (-0.715 - 0.698i)T \)
41 \( 1 + (0.958 - 0.285i)T \)
43 \( 1 + (0.120 + 0.992i)T \)
47 \( 1 + (0.0724 - 0.997i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.958 + 0.285i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.120 + 0.992i)T \)
71 \( 1 + (-0.215 + 0.976i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.485 + 0.873i)T \)
83 \( 1 + (0.989 - 0.144i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.836 + 0.548i)T \)
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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

−20.26645443904763777714197227717, −19.45187137603007664670003361029, −18.9328955479534817372507348275, −18.78219611518304818515771332616, −17.661758317673658831185133227347, −16.70457508081812994835411014328, −15.88698205219813860175948768693, −15.32496267419501324317038779799, −14.75900224745768360358310679011, −13.9200103211250454165421685870, −12.96201258874362012200917283496, −12.14260034493436934908035913303, −10.933950834713248079366083853, −10.508008315532366607938710422738, −9.48475764857203146263588976732, −8.827810465712445293630466868564, −8.156391315314940229037928301475, −7.334213476498280639942965968670, −6.537999287434610270164112220619, −6.16612159453902580874374968754, −4.343998375087478964888644939145, −3.48814427321718567553172976100, −2.65585406445368378402068810470, −1.767922446633180851951263892508, −0.467749976681203422535338386093, 0.63719392466159750743643066756, 1.58636565616612387321570011098, 2.79455817636268358676501628869, 3.47801624305563926179173523416, 4.111274620285571151357904725348, 5.46124319044483861964128935171, 6.827083504752157841609376051632, 7.43131183519351121837091249249, 8.293104392450997388249862699007, 9.0110583526578027476367223435, 9.35776008469484850367870567223, 10.40883199993317063719442677847, 11.06979161524985811011823404879, 12.18048675195228407744069937701, 12.9423485330884739687099716775, 13.280737036647620026990776719582, 14.66049301933020868381398157149, 15.53189568351500346722087916610, 16.160335013413779737376046865157, 16.43835833201076840229971002365, 17.64294918934635121210396016047, 18.54130882986036467028365075171, 19.24499785156197038521238138800, 19.68261006939182307609732463259, 20.39056988329039796189061424068

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