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Properties

Label 1-1441-1441.2-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.168 - 0.985i$
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 + (−0.607 + 0.794i)3-s + (0.215 + 0.976i)4-s + (0.0241 − 0.999i)5-s + (−0.970 + 0.239i)6-s + (−0.926 + 0.377i)7-s + (−0.443 + 0.896i)8-s + (−0.262 − 0.964i)9-s + (0.644 − 0.764i)10-s + (−0.906 − 0.421i)12-s + (0.0724 + 0.997i)13-s + (−0.958 − 0.285i)14-s + (0.779 + 0.626i)15-s + (−0.906 + 0.421i)16-s + (0.120 + 0.992i)17-s + (0.399 − 0.916i)18-s + ⋯
L(s)  = 1  + (0.779 + 0.626i)2-s + (−0.607 + 0.794i)3-s + (0.215 + 0.976i)4-s + (0.0241 − 0.999i)5-s + (−0.970 + 0.239i)6-s + (−0.926 + 0.377i)7-s + (−0.443 + 0.896i)8-s + (−0.262 − 0.964i)9-s + (0.644 − 0.764i)10-s + (−0.906 − 0.421i)12-s + (0.0724 + 0.997i)13-s + (−0.958 − 0.285i)14-s + (0.779 + 0.626i)15-s + (−0.906 + 0.421i)16-s + (0.120 + 0.992i)17-s + (0.399 − 0.916i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.168 - 0.985i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.168 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2083488326 + 0.2471113758i\)
\(L(\frac12)\) \(\approx\) \(-0.2083488326 + 0.2471113758i\)
\(L(1)\) \(\approx\) \(0.7278559806 + 0.5717549941i\)
\(L(1)\) \(\approx\) \(0.7278559806 + 0.5717549941i\)

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 \( 1 + (-0.607 + 0.794i)T \)
5 \( 1 + (0.0241 - 0.999i)T \)
7 \( 1 + (-0.926 + 0.377i)T \)
13 \( 1 + (0.0724 + 0.997i)T \)
17 \( 1 + (0.120 + 0.992i)T \)
19 \( 1 + (-0.906 - 0.421i)T \)
23 \( 1 + (0.443 + 0.896i)T \)
29 \( 1 + (0.836 + 0.548i)T \)
31 \( 1 + (0.443 - 0.896i)T \)
37 \( 1 + (-0.995 - 0.0965i)T \)
41 \( 1 + (-0.120 + 0.992i)T \)
43 \( 1 + (-0.958 + 0.285i)T \)
47 \( 1 + (-0.779 - 0.626i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.485 - 0.873i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.958 - 0.285i)T \)
71 \( 1 + (0.443 - 0.896i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.0241 - 0.999i)T \)
83 \( 1 + (0.995 - 0.0965i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.926 + 0.377i)T \)
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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

−20.09116882127994533132631649897, −19.30166198023940290635898647855, −18.89831530326207460488558627488, −18.14684884285385520723968266996, −17.36288692607687476985881037299, −16.290836258793884464439303446812, −15.5529021489772635471635516233, −14.66811535269806057795031684705, −13.75239364428241444220712924327, −13.406120700094735359956420252448, −12.360675702457002476648976951485, −12.05390097135879608456707069664, −10.80222859855636821288856648265, −10.559993521956681700851984105595, −9.80636532278100678073293784805, −8.39888709257370464486865754315, −7.15173381669214658059706509896, −6.71283163291808529140379705391, −6.01010057436151896774447307467, −5.181293723286376217623923181755, −4.075152047188825545312261655619, −2.93784190471600199409757705322, −2.611091780876580927898655332856, −1.26033662586714618623872141035, −0.100430397491707765442030636226, 1.766599570544908976407892097403, 3.179208139312609814206276148921, 3.9349108630780624846322987487, 4.697779009145158241572927862557, 5.35748411783782692238903007901, 6.309818258153570457668661229530, 6.65954518922219065891443611708, 8.13416626715520107378535784961, 8.90679493926025492845918116116, 9.484388708230332380641773922846, 10.54510466543748432976974484714, 11.68989548728825156789972086973, 12.07927899701202158376658148077, 13.00451912930228678424402303655, 13.46662409510559346895963016402, 14.74963587136527523619603719894, 15.29749640310047956653976431220, 16.07503082135496124138623313809, 16.56453113176654160718277094780, 17.11404161349732300982533472486, 17.83992385540743010157316240900, 19.19397284935867024248779685301, 19.90207105456185814023867869411, 20.953201906023484341507973614095, 21.50515676736007534288366932391

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