Properties

Label 2-1441-1.1-c1-0-45
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  + 0.867·2-s + 0.595·3-s − 1.24·4-s + 3.84·5-s + 0.516·6-s + 1.26·7-s − 2.81·8-s − 2.64·9-s + 3.33·10-s − 11-s − 0.742·12-s + 5.03·13-s + 1.09·14-s + 2.28·15-s + 0.0519·16-s − 3.67·17-s − 2.29·18-s + 3.68·19-s − 4.79·20-s + 0.755·21-s − 0.867·22-s + 5.90·23-s − 1.67·24-s + 9.77·25-s + 4.37·26-s − 3.36·27-s − 1.58·28-s + ⋯
L(s)  = 1  + 0.613·2-s + 0.343·3-s − 0.623·4-s + 1.71·5-s + 0.210·6-s + 0.479·7-s − 0.995·8-s − 0.881·9-s + 1.05·10-s − 0.301·11-s − 0.214·12-s + 1.39·13-s + 0.293·14-s + 0.590·15-s + 0.0129·16-s − 0.890·17-s − 0.540·18-s + 0.845·19-s − 1.07·20-s + 0.164·21-s − 0.184·22-s + 1.23·23-s − 0.342·24-s + 1.95·25-s + 0.857·26-s − 0.646·27-s − 0.298·28-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

\(L(1)\) \(\approx\) \(2.971037199\)
\(L(\frac12)\) \(\approx\) \(2.971037199\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.867T + 2T^{2} \)
3 \( 1 - 0.595T + 3T^{2} \)
5 \( 1 - 3.84T + 5T^{2} \)
7 \( 1 - 1.26T + 7T^{2} \)
13 \( 1 - 5.03T + 13T^{2} \)
17 \( 1 + 3.67T + 17T^{2} \)
19 \( 1 - 3.68T + 19T^{2} \)
23 \( 1 - 5.90T + 23T^{2} \)
29 \( 1 - 8.38T + 29T^{2} \)
31 \( 1 - 4.86T + 31T^{2} \)
37 \( 1 + 2.88T + 37T^{2} \)
41 \( 1 + 1.35T + 41T^{2} \)
43 \( 1 + 7.22T + 43T^{2} \)
47 \( 1 + 2.45T + 47T^{2} \)
53 \( 1 - 2.17T + 53T^{2} \)
59 \( 1 - 9.50T + 59T^{2} \)
61 \( 1 + 3.72T + 61T^{2} \)
67 \( 1 - 4.48T + 67T^{2} \)
71 \( 1 + 4.16T + 71T^{2} \)
73 \( 1 + 7.67T + 73T^{2} \)
79 \( 1 - 5.11T + 79T^{2} \)
83 \( 1 + 4.45T + 83T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 + 2.95T + 97T^{2} \)
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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

−9.449852705612634286908671713648, −8.571222494561394418958432210622, −8.474940725971662482869395839250, −6.72777976163098320292799651421, −6.05593377016985594178631318728, −5.31519655149209342396795116088, −4.72105834835624900616251018583, −3.32072843355866764856951966710, −2.59411014527865336898488329153, −1.24631142996277733658440059584, 1.24631142996277733658440059584, 2.59411014527865336898488329153, 3.32072843355866764856951966710, 4.72105834835624900616251018583, 5.31519655149209342396795116088, 6.05593377016985594178631318728, 6.72777976163098320292799651421, 8.474940725971662482869395839250, 8.571222494561394418958432210622, 9.449852705612634286908671713648

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