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Properties

Label 2-39e2-13.2-c0-0-1
Degree $2$
Conductor $1521$
Sign $0.999 - 0.0386i$
Analytic cond. $0.759077$
Root an. cond. $0.871250$
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  + (−1.36 − 0.366i)2-s + (0.866 + 0.5i)4-s + (−1 + i)5-s + (1.73 − i)10-s + (0.366 − 1.36i)11-s + (−0.499 − 0.866i)16-s + (−1.36 + 0.366i)20-s + (−1 + 1.73i)22-s − i·25-s + (0.366 + 1.36i)32-s + (1.36 + 0.366i)41-s + (1.73 + i)43-s + (1 − 0.999i)44-s + (1 + i)47-s + (−0.866 + 0.5i)49-s + (−0.366 + 1.36i)50-s + ⋯
L(s)  = 1  + (−1.36 − 0.366i)2-s + (0.866 + 0.5i)4-s + (−1 + i)5-s + (1.73 − i)10-s + (0.366 − 1.36i)11-s + (−0.499 − 0.866i)16-s + (−1.36 + 0.366i)20-s + (−1 + 1.73i)22-s − i·25-s + (0.366 + 1.36i)32-s + (1.36 + 0.366i)41-s + (1.73 + i)43-s + (1 − 0.999i)44-s + (1 + i)47-s + (−0.866 + 0.5i)49-s + (−0.366 + 1.36i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.999 - 0.0386i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1432, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ 0.999 - 0.0386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4413868228\)
\(L(\frac12)\) \(\approx\) \(0.4413868228\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
5 \( 1 + (1 - i)T - iT^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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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

−9.517621805985242699266950205914, −8.972028883293363817145065015291, −7.985696200793325042684205832657, −7.69403531890582378433062542675, −6.71747595334367771324026584177, −5.84994662107991887093620573936, −4.39267900832395775358801435636, −3.35961610706580435650370450950, −2.54243219611971349729838532073, −0.910976812824809821657976240477, 0.824396872911944168976828545014, 2.10151424966125618268324803474, 3.96832653364552590580840849863, 4.47262492310976724208332708394, 5.65189500314977577782628096744, 6.99422507541652918191920193203, 7.36161599127269745674835161000, 8.195991541986127494232348791197, 8.806668342244214125187169839931, 9.433887188630957730582729891218

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