Properties

Label 2-1441-1.1-c1-0-19
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  − 1.61·2-s − 1.01·3-s + 0.612·4-s + 4.09·5-s + 1.63·6-s − 3.10·7-s + 2.24·8-s − 1.97·9-s − 6.61·10-s − 11-s − 0.621·12-s − 4.38·13-s + 5.01·14-s − 4.14·15-s − 4.85·16-s − 2.31·17-s + 3.18·18-s + 7.39·19-s + 2.50·20-s + 3.14·21-s + 1.61·22-s − 1.81·23-s − 2.27·24-s + 11.7·25-s + 7.08·26-s + 5.04·27-s − 1.89·28-s + ⋯
L(s)  = 1  − 1.14·2-s − 0.585·3-s + 0.306·4-s + 1.83·5-s + 0.668·6-s − 1.17·7-s + 0.792·8-s − 0.657·9-s − 2.09·10-s − 0.301·11-s − 0.179·12-s − 1.21·13-s + 1.33·14-s − 1.07·15-s − 1.21·16-s − 0.560·17-s + 0.751·18-s + 1.69·19-s + 0.560·20-s + 0.685·21-s + 0.344·22-s − 0.378·23-s − 0.463·24-s + 2.35·25-s + 1.39·26-s + 0.969·27-s − 0.358·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6576510071\)
\(L(\frac12)\) \(\approx\) \(0.6576510071\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.61T + 2T^{2} \)
3 \( 1 + 1.01T + 3T^{2} \)
5 \( 1 - 4.09T + 5T^{2} \)
7 \( 1 + 3.10T + 7T^{2} \)
13 \( 1 + 4.38T + 13T^{2} \)
17 \( 1 + 2.31T + 17T^{2} \)
19 \( 1 - 7.39T + 19T^{2} \)
23 \( 1 + 1.81T + 23T^{2} \)
29 \( 1 + 1.18T + 29T^{2} \)
31 \( 1 - 9.07T + 31T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 - 0.113T + 41T^{2} \)
43 \( 1 + 0.277T + 43T^{2} \)
47 \( 1 + 1.35T + 47T^{2} \)
53 \( 1 + 2.17T + 53T^{2} \)
59 \( 1 - 1.50T + 59T^{2} \)
61 \( 1 - 3.32T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 5.49T + 73T^{2} \)
79 \( 1 + 5.98T + 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 - 4.31T + 89T^{2} \)
97 \( 1 - 6.54T + 97T^{2} \)
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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

−9.622520910710937778406009766374, −9.111845288208408742803009214578, −8.100002596116647845798500207592, −6.96834752469461103793545348036, −6.40741753630123717979483262431, −5.47082058261216323077683089659, −4.89836789034801633061475071164, −3.00652518461785122003669460429, −2.14011629895950778257720148749, −0.69360951520977806801534151486, 0.69360951520977806801534151486, 2.14011629895950778257720148749, 3.00652518461785122003669460429, 4.89836789034801633061475071164, 5.47082058261216323077683089659, 6.40741753630123717979483262431, 6.96834752469461103793545348036, 8.100002596116647845798500207592, 9.111845288208408742803009214578, 9.622520910710937778406009766374

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