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Properties

Label 2-1441-1441.785-c0-0-2
Degree $2$
Conductor $1441$
Sign $0.163 - 0.986i$
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  + (−0.263 + 0.809i)3-s + (0.309 + 0.951i)4-s + (−0.101 + 0.0738i)5-s + (−0.393 − 1.21i)7-s + (0.222 + 0.161i)9-s + (0.728 − 0.684i)11-s − 0.851·12-s + (1.60 + 1.16i)13-s + (−0.0330 − 0.101i)15-s + (−0.809 + 0.587i)16-s + (−0.101 − 0.0738i)20-s + 1.08·21-s + (−0.304 + 0.936i)25-s + (−0.878 + 0.638i)27-s + (1.03 − 0.749i)28-s + ⋯
L(s)  = 1  + (−0.263 + 0.809i)3-s + (0.309 + 0.951i)4-s + (−0.101 + 0.0738i)5-s + (−0.393 − 1.21i)7-s + (0.222 + 0.161i)9-s + (0.728 − 0.684i)11-s − 0.851·12-s + (1.60 + 1.16i)13-s + (−0.0330 − 0.101i)15-s + (−0.809 + 0.587i)16-s + (−0.101 − 0.0738i)20-s + 1.08·21-s + (−0.304 + 0.936i)25-s + (−0.878 + 0.638i)27-s + (1.03 − 0.749i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.163 - 0.986i$
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: \((2,\ 1441,\ (\ :0),\ 0.163 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126565685\)
\(L(\frac12)\) \(\approx\) \(1.126565685\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.728 + 0.684i)T \)
131 \( 1 - T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.263 - 0.809i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.101 - 0.0738i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.393 + 1.21i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.60 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.85T + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.331 - 1.01i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.101 - 0.0738i)T + (0.309 - 0.951i)T^{2} \)
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

−9.949283625812801487671628440389, −9.056094980959580593996994539317, −8.436961783751229518709685526659, −7.31062137107570548370146446671, −6.81476876881380230098992472922, −5.90455184232091137016912332364, −4.49517585564294615637420683599, −3.75660595533889249161798006051, −3.51123157812075346746697627819, −1.60724758205299805399170265567, 1.11117152733352728411108767987, 2.06135219026818999614812527308, 3.31952874847158390091045087929, 4.66055514538297372736396372943, 5.81193957667824407435161898903, 6.20077606041682854935358811494, 6.80261737960299837728642480968, 7.950189157895267060410262410985, 8.741219109127375659064130162835, 9.639703498693218887345302546103

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