Properties

Label 2-1441-1.1-c1-0-49
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  − 1.13·2-s + 3.40·3-s − 0.701·4-s − 0.104·5-s − 3.88·6-s + 1.00·7-s + 3.07·8-s + 8.62·9-s + 0.118·10-s + 11-s − 2.38·12-s + 1.47·13-s − 1.14·14-s − 0.354·15-s − 2.10·16-s − 0.523·17-s − 9.82·18-s + 4.88·19-s + 0.0729·20-s + 3.42·21-s − 1.13·22-s − 3.18·23-s + 10.4·24-s − 4.98·25-s − 1.67·26-s + 19.1·27-s − 0.704·28-s + ⋯
L(s)  = 1  − 0.805·2-s + 1.96·3-s − 0.350·4-s − 0.0465·5-s − 1.58·6-s + 0.379·7-s + 1.08·8-s + 2.87·9-s + 0.0375·10-s + 0.301·11-s − 0.689·12-s + 0.407·13-s − 0.306·14-s − 0.0916·15-s − 0.526·16-s − 0.126·17-s − 2.31·18-s + 1.12·19-s + 0.0163·20-s + 0.747·21-s − 0.242·22-s − 0.664·23-s + 2.14·24-s − 0.997·25-s − 0.328·26-s + 3.68·27-s − 0.133·28-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

\(L(1)\) \(\approx\) \(2.261536586\)
\(L(\frac12)\) \(\approx\) \(2.261536586\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.13T + 2T^{2} \)
3 \( 1 - 3.40T + 3T^{2} \)
5 \( 1 + 0.104T + 5T^{2} \)
7 \( 1 - 1.00T + 7T^{2} \)
13 \( 1 - 1.47T + 13T^{2} \)
17 \( 1 + 0.523T + 17T^{2} \)
19 \( 1 - 4.88T + 19T^{2} \)
23 \( 1 + 3.18T + 23T^{2} \)
29 \( 1 - 4.16T + 29T^{2} \)
31 \( 1 + 7.69T + 31T^{2} \)
37 \( 1 - 4.51T + 37T^{2} \)
41 \( 1 + 4.28T + 41T^{2} \)
43 \( 1 + 7.09T + 43T^{2} \)
47 \( 1 - 7.07T + 47T^{2} \)
53 \( 1 + 4.44T + 53T^{2} \)
59 \( 1 - 3.97T + 59T^{2} \)
61 \( 1 - 14.0T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 7.92T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 1.05T + 83T^{2} \)
89 \( 1 + 9.63T + 89T^{2} \)
97 \( 1 + 3.03T + 97T^{2} \)
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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

−9.460790774305062761429072717117, −8.694171022335246543088809236986, −8.075926197210236523570904808139, −7.64866106798029036339091251048, −6.74999408899245658731874014158, −5.12678217275312640216277672164, −4.07774449682831113423797590890, −3.48895013221028873257927561235, −2.17008746738491976649550398286, −1.28357232884682650816241017473, 1.28357232884682650816241017473, 2.17008746738491976649550398286, 3.48895013221028873257927561235, 4.07774449682831113423797590890, 5.12678217275312640216277672164, 6.74999408899245658731874014158, 7.64866106798029036339091251048, 8.075926197210236523570904808139, 8.694171022335246543088809236986, 9.460790774305062761429072717117

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