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Properties

Label 2-39e2-13.6-c0-0-0
Degree $2$
Conductor $1521$
Sign $-0.852 + 0.522i$
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  + (0.366 + 1.36i)2-s + (−0.866 + 0.5i)4-s + (−1 + i)5-s + (−1.73 − i)10-s + (−1.36 + 0.366i)11-s + (−0.499 + 0.866i)16-s + (0.366 − 1.36i)20-s + (−1 − 1.73i)22-s − i·25-s + (−1.36 − 0.366i)32-s + (−0.366 − 1.36i)41-s + (−1.73 + i)43-s + (0.999 − i)44-s + (1 + i)47-s + (0.866 + 0.5i)49-s + (1.36 − 0.366i)50-s + ⋯
L(s)  = 1  + (0.366 + 1.36i)2-s + (−0.866 + 0.5i)4-s + (−1 + i)5-s + (−1.73 − i)10-s + (−1.36 + 0.366i)11-s + (−0.499 + 0.866i)16-s + (0.366 − 1.36i)20-s + (−1 − 1.73i)22-s − i·25-s + (−1.36 − 0.366i)32-s + (−0.366 − 1.36i)41-s + (−1.73 + i)43-s + (0.999 − i)44-s + (1 + i)47-s + (0.866 + 0.5i)49-s + (1.36 − 0.366i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.852 + 0.522i$
Analytic conductor: \(0.759077\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :0),\ -0.852 + 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8264134670\)
\(L(\frac12)\) \(\approx\) \(0.8264134670\)
\(L(1)\) 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 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
5 \( 1 + (1 - i)T - iT^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
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

−10.30793376712768423479968577389, −8.997958330881478425091996714051, −8.039360190474106441684988036623, −7.59316547239000872640145927696, −7.03367002451891164136444409618, −6.20659325994628558139793537003, −5.29686357523040008227032431144, −4.49944076069812045372961427709, −3.49437219657911600204011765774, −2.39672333993709651934274194279, 0.56916806197531090722533623677, 2.00310205356506553350699916122, 3.13700057190234228982555463008, 3.89602411932736342681797380301, 4.83321405960523162610136511806, 5.35234724367793009639864270967, 6.86514578702649269646091878782, 7.903307555225814362330554798246, 8.393771341284142566616043789288, 9.363276887210903784860466108718

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