Properties

Label 2-143e2-1.1-c1-0-0
Degree $2$
Conductor $20449$
Sign $1$
Analytic cond. $163.286$
Root an. cond. $12.7783$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s − 4-s − 5-s − 2·6-s + 2·7-s + 3·8-s + 9-s + 10-s − 2·12-s − 2·14-s − 2·15-s − 16-s − 5·17-s − 18-s − 6·19-s + 20-s + 4·21-s + 2·23-s + 6·24-s − 4·25-s − 4·27-s − 2·28-s + 9·29-s + 2·30-s + 2·31-s − 5·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.577·12-s − 0.534·14-s − 0.516·15-s − 1/4·16-s − 1.21·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s + 0.872·21-s + 0.417·23-s + 1.22·24-s − 4/5·25-s − 0.769·27-s − 0.377·28-s + 1.67·29-s + 0.365·30-s + 0.359·31-s − 0.883·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20449\)    =    \(11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(163.286\)
Root analytic conductor: \(12.7783\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 20449,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.386326356\)
\(L(\frac12)\) \(\approx\) \(1.386326356\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−15.63392990132735, −14.93971239576520, −14.61098879250781, −14.04622754894850, −13.51326250038537, −13.10719487189147, −12.47939003026753, −11.55128700141921, −11.23985138453662, −10.41615178169868, −10.08824259735816, −9.160700204340424, −8.853182083162407, −8.403789255692855, −7.978839415719147, −7.478582847095663, −6.705310773959716, −5.940586387554647, −4.897340382022149, −4.353877615934425, −4.023761559289007, −2.984646901225348, −2.276775870831780, −1.605759713574050, −0.5111797938228253, 0.5111797938228253, 1.605759713574050, 2.276775870831780, 2.984646901225348, 4.023761559289007, 4.353877615934425, 4.897340382022149, 5.940586387554647, 6.705310773959716, 7.478582847095663, 7.978839415719147, 8.403789255692855, 8.853182083162407, 9.160700204340424, 10.08824259735816, 10.41615178169868, 11.23985138453662, 11.55128700141921, 12.47939003026753, 13.10719487189147, 13.51326250038537, 14.04622754894850, 14.61098879250781, 14.93971239576520, 15.63392990132735

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