Properties

Label 2-39e2-1.1-c3-0-98
Degree $2$
Conductor $1521$
Sign $-1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s − 31.1·7-s + 64·16-s + 155.·19-s − 125·25-s + 249.·28-s − 155.·31-s + 436.·37-s + 520·43-s + 629·49-s − 182·61-s − 512·64-s − 654.·67-s + 374.·73-s − 1.24e3·76-s − 884·79-s − 1.37e3·97-s + 1.00e3·100-s − 1.82e3·103-s + 2.18e3·109-s − 1.99e3·112-s + ⋯
L(s)  = 1  − 4-s − 1.68·7-s + 16-s + 1.88·19-s − 25-s + 1.68·28-s − 0.903·31-s + 1.93·37-s + 1.84·43-s + 1.83·49-s − 0.382·61-s − 64-s − 1.19·67-s + 0.599·73-s − 1.88·76-s − 1.25·79-s − 1.43·97-s + 100-s − 1.74·103-s + 1.91·109-s − 1.68·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(89.7419\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 8T^{2} \)
5 \( 1 + 125T^{2} \)
7 \( 1 + 31.1T + 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 155.T + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 + 155.T + 2.97e4T^{2} \)
37 \( 1 - 436.T + 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 520T + 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 182T + 2.26e5T^{2} \)
67 \( 1 + 654.T + 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 374.T + 3.89e5T^{2} \)
79 \( 1 + 884T + 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + 1.37e3T + 9.12e5T^{2} \)
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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

−8.998214003713661364437147075987, −7.83327389115685696044828067506, −7.21079640178592173931393431180, −6.01113100590954679548542440306, −5.57895385905292561629522994632, −4.33717937869256377580484571026, −3.55505660731407155280105406811, −2.77187807113650451896034189970, −1.00655724279934310474327491070, 0, 1.00655724279934310474327491070, 2.77187807113650451896034189970, 3.55505660731407155280105406811, 4.33717937869256377580484571026, 5.57895385905292561629522994632, 6.01113100590954679548542440306, 7.21079640178592173931393431180, 7.83327389115685696044828067506, 8.998214003713661364437147075987

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