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Properties

Label 1-1441-1441.384-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.498 - 0.867i$
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 + (0.0241 − 0.999i)3-s + (0.215 + 0.976i)4-s + (0.958 − 0.285i)5-s + (0.644 − 0.764i)6-s + (0.527 − 0.849i)7-s + (−0.443 + 0.896i)8-s + (−0.998 − 0.0483i)9-s + (0.926 + 0.377i)10-s + (0.981 − 0.192i)12-s + (−0.926 + 0.377i)13-s + (0.943 − 0.331i)14-s + (−0.262 − 0.964i)15-s + (−0.906 + 0.421i)16-s + (−0.681 − 0.732i)17-s + (−0.748 − 0.663i)18-s + ⋯
L(s)  = 1  + (0.779 + 0.626i)2-s + (0.0241 − 0.999i)3-s + (0.215 + 0.976i)4-s + (0.958 − 0.285i)5-s + (0.644 − 0.764i)6-s + (0.527 − 0.849i)7-s + (−0.443 + 0.896i)8-s + (−0.998 − 0.0483i)9-s + (0.926 + 0.377i)10-s + (0.981 − 0.192i)12-s + (−0.926 + 0.377i)13-s + (0.943 − 0.331i)14-s + (−0.262 − 0.964i)15-s + (−0.906 + 0.421i)16-s + (−0.681 − 0.732i)17-s + (−0.748 − 0.663i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.498 - 0.867i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (384, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.498 - 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.297299464 - 1.329758824i\)
\(L(\frac12)\) \(\approx\) \(2.297299464 - 1.329758824i\)
\(L(1)\) \(\approx\) \(1.760920619 - 0.2586270830i\)
\(L(1)\) \(\approx\) \(1.760920619 - 0.2586270830i\)

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 \( 1 + (0.0241 - 0.999i)T \)
5 \( 1 + (0.958 - 0.285i)T \)
7 \( 1 + (0.527 - 0.849i)T \)
13 \( 1 + (-0.926 + 0.377i)T \)
17 \( 1 + (-0.681 - 0.732i)T \)
19 \( 1 + (0.120 - 0.992i)T \)
23 \( 1 + (0.989 - 0.144i)T \)
29 \( 1 + (-0.998 + 0.0483i)T \)
31 \( 1 + (-0.715 - 0.698i)T \)
37 \( 1 + (0.861 - 0.506i)T \)
41 \( 1 + (0.906 + 0.421i)T \)
43 \( 1 + (0.607 - 0.794i)T \)
47 \( 1 + (0.354 - 0.935i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.485 - 0.873i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.607 + 0.794i)T \)
71 \( 1 + (-0.885 + 0.464i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (0.399 + 0.916i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.0724 + 0.997i)T \)
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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

−21.08774827678595339089743061499, −20.432643837097121723262166117421, −19.55024693969748971320851253479, −18.72566417587510568092467020258, −17.83659637895444002815949719743, −17.13201310989403640675123933324, −16.1434098855711936566317789073, −15.15984644098921198255906396425, −14.708308848836807371122142319393, −14.2993805895076659475142229506, −13.15820149446944777605093519921, −12.5109157298865337852325956832, −11.53382854778978996611907213768, −10.82130388344707246148327501706, −10.251313926152922781064465213, −9.360857270080130172465202552218, −8.914220358610930569735370860934, −7.512310683638616367251068187774, −6.12850252629537132035480843844, −5.66628745373119650155977885720, −4.96805310566784768921410480648, −4.14865783170207592453449391301, −3.01145873833461944938495493693, −2.44063341077348392971893278606, −1.50432329828105353234949736042, 0.71828307101066878892916193461, 2.129736186470951850293779323542, 2.55293128438039699860781589928, 3.95771541635817517603092533298, 5.00247223685505079148129107710, 5.45275725167442643143115110325, 6.65742580115273181524299372496, 7.06661017116201800468332998623, 7.75255239118748676200516508934, 8.83991323613019807195317021097, 9.47582741827880148408591251489, 11.01018427491212163816884061286, 11.46803585175119557549370788580, 12.57360999572745297338922792846, 13.23158271152486257985790751527, 13.58758317446943142310543263456, 14.47260413823532134204076484979, 14.86740978743047091106756743990, 16.29900617515931169550308430937, 16.90604891153884118908680052948, 17.55797075752005620564796326286, 17.92830527520487160771382403163, 19.04759557993844415908295376452, 20.25286064446148865983455227678, 20.42736464043551489282408522825

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