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Properties

Label 1-1441-1441.483-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.970 + 0.242i$
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  + (0.981 − 0.192i)2-s + (0.215 − 0.976i)3-s + (0.926 − 0.377i)4-s + (−0.861 − 0.506i)5-s + (0.0241 − 0.999i)6-s + (−0.958 − 0.285i)7-s + (0.836 − 0.548i)8-s + (−0.906 − 0.421i)9-s + (−0.943 − 0.331i)10-s + (−0.168 − 0.985i)12-s + (0.943 − 0.331i)13-s + (−0.995 − 0.0965i)14-s + (−0.681 + 0.732i)15-s + (0.715 − 0.698i)16-s + (−0.443 − 0.896i)17-s + (−0.970 − 0.239i)18-s + ⋯
L(s)  = 1  + (0.981 − 0.192i)2-s + (0.215 − 0.976i)3-s + (0.926 − 0.377i)4-s + (−0.861 − 0.506i)5-s + (0.0241 − 0.999i)6-s + (−0.958 − 0.285i)7-s + (0.836 − 0.548i)8-s + (−0.906 − 0.421i)9-s + (−0.943 − 0.331i)10-s + (−0.168 − 0.985i)12-s + (0.943 − 0.331i)13-s + (−0.995 − 0.0965i)14-s + (−0.681 + 0.732i)15-s + (0.715 − 0.698i)16-s + (−0.443 − 0.896i)17-s + (−0.970 − 0.239i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.970 + 0.242i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2357475566 - 1.913101722i\)
\(L(\frac12)\) \(\approx\) \(-0.2357475566 - 1.913101722i\)
\(L(1)\) \(\approx\) \(1.077278470 - 1.071795179i\)
\(L(1)\) \(\approx\) \(1.077278470 - 1.071795179i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.981 - 0.192i)T \)
3 \( 1 + (0.215 - 0.976i)T \)
5 \( 1 + (-0.861 - 0.506i)T \)
7 \( 1 + (-0.958 - 0.285i)T \)
13 \( 1 + (0.943 - 0.331i)T \)
17 \( 1 + (-0.443 - 0.896i)T \)
19 \( 1 + (0.885 - 0.464i)T \)
23 \( 1 + (0.262 - 0.964i)T \)
29 \( 1 + (-0.906 + 0.421i)T \)
31 \( 1 + (-0.779 - 0.626i)T \)
37 \( 1 + (0.0724 + 0.997i)T \)
41 \( 1 + (-0.715 - 0.698i)T \)
43 \( 1 + (-0.399 - 0.916i)T \)
47 \( 1 + (-0.120 + 0.992i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.989 + 0.144i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.399 + 0.916i)T \)
71 \( 1 + (0.354 - 0.935i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.748 + 0.663i)T \)
83 \( 1 + (-0.527 - 0.849i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.607 + 0.794i)T \)
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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

−21.45664938981758702569558350397, −20.39900203275715680944169890999, −19.82159042876161354775480811156, −19.22907149307375176784942355781, −18.19375392799818408613322122552, −16.894037916201206062246001219635, −16.191588043294849851037479541405, −15.78734924685867238374796438983, −15.10756936263441867894744303655, −14.52666132318758344417734830768, −13.577801692492283176566483734541, −12.89874326967345893653968492829, −11.832879382622402808999379454022, −11.26436792056174023478886585764, −10.57942954459612686579700465773, −9.62725879257375496358227779359, −8.639537756286834910192957219655, −7.81056337789551859655845740753, −6.84715846029846307148518550813, −6.02568484767843811631249901374, −5.28174981710893426073397529032, −4.126946581529002086770325790972, −3.53182358682344917140905501435, −3.17847797896670685977667963668, −1.89896032038272828347466704937, 0.48959476262581012691310178744, 1.42532089203715973099596460940, 2.76562508441321632776067668930, 3.3257829947953089556808526847, 4.188669961798295526843506745787, 5.28116570901699695596102508401, 6.10126407164919085466579872281, 7.06896741579824096699558978550, 7.38784682945396208167279782839, 8.560631036913796325097632292319, 9.35265240923988076429652780259, 10.64555923374631231429007760237, 11.44319288309749691100495461458, 12.03774769506611001684632490157, 12.88995970089023173610092597184, 13.27254359077188514061531193859, 13.93548766144481128746478798760, 14.975302026216712529958730642684, 15.72186058528775810401204597932, 16.33355007590145277954001320976, 17.111047333138259527478394890894, 18.58824871104846560649031981678, 18.77021065819623432960922227629, 19.9094633561834319519058012591, 20.29408549290201945910256967460

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