Properties

Label 2-1441-1441.785-c0-0-1
Degree $2$
Conductor $1441$
Sign $-0.887 - 0.460i$
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  + (−0.613 + 1.88i)3-s + (0.309 + 0.951i)4-s + (1.03 − 0.749i)5-s + (0.598 + 1.84i)7-s + (−2.37 − 1.72i)9-s + (−0.425 − 0.904i)11-s − 1.98·12-s + (0.303 + 0.220i)13-s + (0.781 + 2.40i)15-s + (−0.809 + 0.587i)16-s + (1.03 + 0.749i)20-s − 3.84·21-s + (0.193 − 0.594i)25-s + (3.10 − 2.25i)27-s + (−1.56 + 1.13i)28-s + ⋯
L(s)  = 1  + (−0.613 + 1.88i)3-s + (0.309 + 0.951i)4-s + (1.03 − 0.749i)5-s + (0.598 + 1.84i)7-s + (−2.37 − 1.72i)9-s + (−0.425 − 0.904i)11-s − 1.98·12-s + (0.303 + 0.220i)13-s + (0.781 + 2.40i)15-s + (−0.809 + 0.587i)16-s + (1.03 + 0.749i)20-s − 3.84·21-s + (0.193 − 0.594i)25-s + (3.10 − 2.25i)27-s + (−1.56 + 1.13i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.887 - 0.460i$
Analytic conductor: \(0.719152\)
Root analytic conductor: \(0.848028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :0),\ -0.887 - 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.111074086\)
\(L(\frac12)\) \(\approx\) \(1.111074086\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.425 + 0.904i)T \)
131 \( 1 - T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (-1.03 + 0.749i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.598 - 1.84i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.303 - 0.220i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.450 + 1.38i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.07T + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.0388 - 0.119i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-1.03 + 0.749i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−9.914929781500349179600604120876, −9.059051040029955153473914798733, −8.849242634853497799885374254208, −8.175980446973584879937408003373, −6.29605222750459110748903808287, −5.62084046717936386133323286614, −5.25002080807157209692250930165, −4.27496882544134104259520021538, −3.18257998736867780663865013726, −2.29300372664594227228657079201, 1.03024471473904774924360263067, 1.77950587063949820314605008006, 2.66392466326180335927685706358, 4.63591013335201917488151613379, 5.58750834094308254453725581054, 6.29552528913398755438474902777, 6.96850885805695072825916070077, 7.36293984421791203779776273709, 8.161533452882893072531014436272, 9.710182705954584309205766246055

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