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Properties

Label 2-1441-1441.916-c0-0-1
Degree $2$
Conductor $1441$
Sign $0.988 + 0.148i$
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  + (0.450 + 1.38i)3-s + (0.309 − 0.951i)4-s + (−0.866 − 0.629i)5-s + (0.0388 − 0.119i)7-s + (−0.910 + 0.661i)9-s + (0.876 − 0.481i)11-s + 1.45·12-s + (0.688 − 0.500i)13-s + (0.482 − 1.48i)15-s + (−0.809 − 0.587i)16-s + (−0.866 + 0.629i)20-s + 0.183·21-s + (0.0458 + 0.141i)25-s + (−0.148 − 0.107i)27-s + (−0.101 − 0.0738i)28-s + ⋯
L(s)  = 1  + (0.450 + 1.38i)3-s + (0.309 − 0.951i)4-s + (−0.866 − 0.629i)5-s + (0.0388 − 0.119i)7-s + (−0.910 + 0.661i)9-s + (0.876 − 0.481i)11-s + 1.45·12-s + (0.688 − 0.500i)13-s + (0.482 − 1.48i)15-s + (−0.809 − 0.587i)16-s + (−0.866 + 0.629i)20-s + 0.183·21-s + (0.0458 + 0.141i)25-s + (−0.148 − 0.107i)27-s + (−0.101 − 0.0738i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.988 + 0.148i$
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: \((2,\ 1441,\ (\ :0),\ 0.988 + 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.236574814\)
\(L(\frac12)\) \(\approx\) \(1.236574814\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.876 + 0.481i)T \)
131 \( 1 - T \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.866 + 0.629i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.0388 + 0.119i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.688 + 0.500i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (0.115 + 0.356i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.93T + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.574 - 1.76i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.866 + 0.629i)T + (0.309 + 0.951i)T^{2} \)
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

−9.580259923864516423022667016444, −9.048984956995513483200909595013, −8.417226463755459159253266039607, −7.40438601401775813208167800281, −6.20029256680256205240368208765, −5.44910696305104589864071686409, −4.40726590638176435793174447201, −3.97786415092820127181403263400, −2.88046755074379016906650741098, −1.11721299176090846084419932670, 1.59425175736476051072354290947, 2.62868416116225390004350528640, 3.55678977908372330755360759176, 4.31923123351299490106231043210, 6.11204283772777489031555445282, 6.86109725515458779830432669803, 7.27803117905320219809858663252, 7.975835542717261791058243860848, 8.611541943485471284743059171884, 9.437090611335515177866054918893

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