Properties

Label 2-1441-1.1-c1-0-41
Degree $2$
Conductor $1441$
Sign $1$
Analytic cond. $11.5064$
Root an. cond. $3.39211$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.11·2-s − 3.18·3-s + 2.49·4-s + 1.61·5-s − 6.74·6-s + 3.36·7-s + 1.03·8-s + 7.11·9-s + 3.42·10-s + 11-s − 7.92·12-s + 0.557·13-s + 7.12·14-s − 5.13·15-s − 2.77·16-s − 3.12·17-s + 15.0·18-s − 1.99·19-s + 4.02·20-s − 10.6·21-s + 2.11·22-s + 9.11·23-s − 3.30·24-s − 2.39·25-s + 1.18·26-s − 13.1·27-s + 8.37·28-s + ⋯
L(s)  = 1  + 1.49·2-s − 1.83·3-s + 1.24·4-s + 0.722·5-s − 2.75·6-s + 1.27·7-s + 0.367·8-s + 2.37·9-s + 1.08·10-s + 0.301·11-s − 2.28·12-s + 0.154·13-s + 1.90·14-s − 1.32·15-s − 0.694·16-s − 0.757·17-s + 3.55·18-s − 0.457·19-s + 0.899·20-s − 2.33·21-s + 0.451·22-s + 1.90·23-s − 0.674·24-s − 0.478·25-s + 0.231·26-s − 2.52·27-s + 1.58·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $1$
Analytic conductor: \(11.5064\)
Root analytic conductor: \(3.39211\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.798795968\)
\(L(\frac12)\) \(\approx\) \(2.798795968\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T \)
131 \( 1 + T \)
good2 \( 1 - 2.11T + 2T^{2} \)
3 \( 1 + 3.18T + 3T^{2} \)
5 \( 1 - 1.61T + 5T^{2} \)
7 \( 1 - 3.36T + 7T^{2} \)
13 \( 1 - 0.557T + 13T^{2} \)
17 \( 1 + 3.12T + 17T^{2} \)
19 \( 1 + 1.99T + 19T^{2} \)
23 \( 1 - 9.11T + 23T^{2} \)
29 \( 1 - 2.14T + 29T^{2} \)
31 \( 1 - 7.14T + 31T^{2} \)
37 \( 1 - 4.22T + 37T^{2} \)
41 \( 1 - 4.84T + 41T^{2} \)
43 \( 1 - 2.12T + 43T^{2} \)
47 \( 1 - 2.67T + 47T^{2} \)
53 \( 1 + 2.57T + 53T^{2} \)
59 \( 1 + 7.07T + 59T^{2} \)
61 \( 1 - 8.43T + 61T^{2} \)
67 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 8.23T + 73T^{2} \)
79 \( 1 - 5.87T + 79T^{2} \)
83 \( 1 + 5.87T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 6.22T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.850695003968239577531280025455, −8.775515648574968951049125613940, −7.40885809980525503367062775977, −6.48547136375272626679256661760, −6.10384111682753211897796683203, −5.15955173394583661504603438144, −4.80360319551024063046686652647, −4.07798715773448782862557019707, −2.38841640829164104578193062044, −1.15294663862356821733989532147, 1.15294663862356821733989532147, 2.38841640829164104578193062044, 4.07798715773448782862557019707, 4.80360319551024063046686652647, 5.15955173394583661504603438144, 6.10384111682753211897796683203, 6.48547136375272626679256661760, 7.40885809980525503367062775977, 8.775515648574968951049125613940, 9.850695003968239577531280025455

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