Properties

Label 2-1441-1.1-c1-0-33
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  − 1.01·2-s − 1.33·3-s − 0.975·4-s + 3.44·5-s + 1.34·6-s + 1.50·7-s + 3.01·8-s − 1.22·9-s − 3.49·10-s + 11-s + 1.29·12-s + 5.15·13-s − 1.52·14-s − 4.59·15-s − 1.09·16-s + 3.32·17-s + 1.24·18-s − 1.84·19-s − 3.36·20-s − 2.00·21-s − 1.01·22-s + 5.95·23-s − 4.01·24-s + 6.89·25-s − 5.21·26-s + 5.62·27-s − 1.46·28-s + ⋯
L(s)  = 1  − 0.715·2-s − 0.769·3-s − 0.487·4-s + 1.54·5-s + 0.550·6-s + 0.568·7-s + 1.06·8-s − 0.408·9-s − 1.10·10-s + 0.301·11-s + 0.375·12-s + 1.42·13-s − 0.406·14-s − 1.18·15-s − 0.274·16-s + 0.807·17-s + 0.292·18-s − 0.423·19-s − 0.752·20-s − 0.436·21-s − 0.215·22-s + 1.24·23-s − 0.819·24-s + 1.37·25-s − 1.02·26-s + 1.08·27-s − 0.276·28-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

\(L(1)\) \(\approx\) \(1.165952488\)
\(L(\frac12)\) \(\approx\) \(1.165952488\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.01T + 2T^{2} \)
3 \( 1 + 1.33T + 3T^{2} \)
5 \( 1 - 3.44T + 5T^{2} \)
7 \( 1 - 1.50T + 7T^{2} \)
13 \( 1 - 5.15T + 13T^{2} \)
17 \( 1 - 3.32T + 17T^{2} \)
19 \( 1 + 1.84T + 19T^{2} \)
23 \( 1 - 5.95T + 23T^{2} \)
29 \( 1 + 4.75T + 29T^{2} \)
31 \( 1 - 1.13T + 31T^{2} \)
37 \( 1 + 6.93T + 37T^{2} \)
41 \( 1 + 12.7T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + 0.187T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 9.24T + 59T^{2} \)
61 \( 1 + 7.30T + 61T^{2} \)
67 \( 1 - 7.00T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 3.06T + 73T^{2} \)
79 \( 1 - 4.22T + 79T^{2} \)
83 \( 1 + 2.76T + 83T^{2} \)
89 \( 1 - 2.67T + 89T^{2} \)
97 \( 1 - 2.37T + 97T^{2} \)
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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

−9.415549152850947535103089228465, −8.861473019096130702326029033598, −8.245540056750322173425202861072, −7.02583506310523933105319433149, −6.10417458860214313992859027749, −5.48153563403949546377909908904, −4.81682389895439939964907394666, −3.44254656754891460534668919223, −1.83217526673657451180109686148, −0.986048953168040130617673845101, 0.986048953168040130617673845101, 1.83217526673657451180109686148, 3.44254656754891460534668919223, 4.81682389895439939964907394666, 5.48153563403949546377909908904, 6.10417458860214313992859027749, 7.02583506310523933105319433149, 8.245540056750322173425202861072, 8.861473019096130702326029033598, 9.415549152850947535103089228465

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