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Properties

Label 2-1441-1441.654-c0-0-0
Degree $2$
Conductor $1441$
Sign $0.204 + 0.978i$
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  + (−1.56 − 1.13i)3-s + (−0.809 + 0.587i)4-s + (−0.115 + 0.356i)5-s + (−1.41 + 1.03i)7-s + (0.850 + 2.61i)9-s + (−0.637 + 0.770i)11-s + 1.93·12-s + (−0.574 − 1.76i)13-s + (0.587 − 0.426i)15-s + (0.309 − 0.951i)16-s + (−0.115 − 0.356i)20-s + 3.39·21-s + (0.695 + 0.505i)25-s + (1.04 − 3.22i)27-s + (0.541 − 1.66i)28-s + ⋯
L(s)  = 1  + (−1.56 − 1.13i)3-s + (−0.809 + 0.587i)4-s + (−0.115 + 0.356i)5-s + (−1.41 + 1.03i)7-s + (0.850 + 2.61i)9-s + (−0.637 + 0.770i)11-s + 1.93·12-s + (−0.574 − 1.76i)13-s + (0.587 − 0.426i)15-s + (0.309 − 0.951i)16-s + (−0.115 − 0.356i)20-s + 3.39·21-s + (0.695 + 0.505i)25-s + (1.04 − 3.22i)27-s + (0.541 − 1.66i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(0.719152\)
Root analytic conductor: \(0.848028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :0),\ 0.204 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2179491234\)
\(L(\frac12)\) \(\approx\) \(0.2179491234\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.637 - 0.770i)T \)
131 \( 1 - T \)
good2 \( 1 + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (1.56 + 1.13i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.115 - 0.356i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.574 + 1.76i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.851T + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.115 - 0.356i)T + (-0.809 - 0.587i)T^{2} \)
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

−9.896518722428637965914907573483, −8.546382343169529529055676620535, −7.71693485926792827525123226115, −7.08013764134113108395180755350, −6.28615398368061793996257473865, −5.32823353947268133940833204047, −5.04388998970704354975027640076, −3.30406044420314732338103994316, −2.39352856895565827197144047126, −0.32571333603065004869830536085, 0.826779254878308605043999028751, 3.44236671547274960706041305740, 4.29678981210342129379024721842, 4.72790461866159690583219965388, 5.72229744838239414593132781158, 6.40508536315507371571552928160, 7.06331892888241960826836219790, 8.760524552252535173773888251948, 9.419294022738848449914696028349, 9.981665451377410471880014751113

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