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Properties

Label 2-1441-1441.130-c0-0-1
Degree $2$
Conductor $1441$
Sign $-0.867 - 0.496i$
Analytic cond. $0.719152$
Root an. cond. $0.848028$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.629i)3-s + (−0.809 − 0.587i)4-s + (0.450 + 1.38i)5-s + (0.688 + 0.500i)7-s + (0.0458 − 0.141i)9-s + (−0.929 + 0.368i)11-s + 1.07·12-s + (0.0388 − 0.119i)13-s + (−1.26 − 0.918i)15-s + (0.309 + 0.951i)16-s + (0.450 − 1.38i)20-s − 0.912·21-s + (−0.910 + 0.661i)25-s + (−0.282 − 0.867i)27-s + (−0.263 − 0.809i)28-s + ⋯
L(s)  = 1  + (−0.866 + 0.629i)3-s + (−0.809 − 0.587i)4-s + (0.450 + 1.38i)5-s + (0.688 + 0.500i)7-s + (0.0458 − 0.141i)9-s + (−0.929 + 0.368i)11-s + 1.07·12-s + (0.0388 − 0.119i)13-s + (−1.26 − 0.918i)15-s + (0.309 + 0.951i)16-s + (0.450 − 1.38i)20-s − 0.912·21-s + (−0.910 + 0.661i)25-s + (−0.282 − 0.867i)27-s + (−0.263 − 0.809i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.867 - 0.496i$
Analytic conductor: \(0.719152\)
Root analytic conductor: \(0.848028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :0),\ -0.867 - 0.496i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5499222234\)
\(L(\frac12)\) \(\approx\) \(0.5499222234\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.929 - 0.368i)T \)
131 \( 1 - T \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.688 - 0.500i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.0388 + 0.119i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (1.56 - 1.13i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 0.374T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (1.41 + 1.03i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−10.18000446120331721561035005814, −9.629396384971341404919450016743, −8.504324660796337968472680190310, −7.67526770834351133218602082863, −6.51764967677670738920110048052, −5.78813593075159208471478050541, −5.12667508124424244962550637138, −4.48766869197673660594364965296, −3.10150673858408065858452965553, −1.93637927423437068477113074040, 0.51579333842829241299304846383, 1.69996178684930393164765660168, 3.45465511008350309676763019530, 4.67506677178535998569519120194, 5.11867386453204513617533597446, 5.85826070703556586363046657239, 7.03755868172233595166493831300, 7.938894252590112796198161173753, 8.501680535792569116276014352121, 9.226500018566236359124878903505

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