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Properties

Label 2-1441-1441.916-c0-0-4
Degree $2$
Conductor $1441$
Sign $-0.712 - 0.701i$
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  + (−0.115 − 0.356i)3-s + (0.309 − 0.951i)4-s + (−1.56 − 1.13i)5-s + (−0.574 + 1.76i)7-s + (0.695 − 0.505i)9-s + (−0.992 − 0.125i)11-s − 0.374·12-s + (−1.41 + 1.03i)13-s + (−0.224 + 0.690i)15-s + (−0.809 − 0.587i)16-s + (−1.56 + 1.13i)20-s + 0.696·21-s + (0.850 + 2.61i)25-s + (−0.563 − 0.409i)27-s + (1.50 + 1.09i)28-s + ⋯
L(s)  = 1  + (−0.115 − 0.356i)3-s + (0.309 − 0.951i)4-s + (−1.56 − 1.13i)5-s + (−0.574 + 1.76i)7-s + (0.695 − 0.505i)9-s + (−0.992 − 0.125i)11-s − 0.374·12-s + (−1.41 + 1.03i)13-s + (−0.224 + 0.690i)15-s + (−0.809 − 0.587i)16-s + (−1.56 + 1.13i)20-s + 0.696·21-s + (0.850 + 2.61i)25-s + (−0.563 − 0.409i)27-s + (1.50 + 1.09i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(0.719152\)
Root analytic conductor: \(0.848028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (916, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1441,\ (\ :0),\ -0.712 - 0.701i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07169405373\)
\(L(\frac12)\) \(\approx\) \(0.07169405373\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.992 + 0.125i)T \)
131 \( 1 - T \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.115 + 0.356i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (1.56 + 1.13i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.574 - 1.76i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (1.41 - 1.03i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 0.125T + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.393 - 1.21i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (1.56 + 1.13i)T + (0.309 + 0.951i)T^{2} \)
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

−9.290528639862277473262758726542, −8.512729156072213607958417391806, −7.54298375797258606809128638820, −6.86649212756511303655286496237, −5.79971771133403065436492706187, −5.05757892790510676526717659426, −4.36709202975875904729080981900, −2.89736579369161970716709029038, −1.76504212503075879670923407166, −0.05563377149387688622689255639, 2.73235762509476184452361648995, 3.35548324144925749939161635388, 4.17310326602568559515420925175, 4.79997720920139552933541322297, 6.65985542107144113337333118465, 7.25379629443674976525902462117, 7.74064700777358857227671538438, 7.979582589533618693811447040303, 9.789481743408217651888624791408, 10.49160434526182870013487303933

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