Properties

Label 2-39e2-1.1-c3-0-182
Degree $2$
Conductor $1521$
Sign $-1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 8·4-s + 17·5-s − 20·7-s + 68·10-s − 32·11-s − 80·14-s − 64·16-s + 13·17-s − 30·19-s + 136·20-s − 128·22-s − 78·23-s + 164·25-s − 160·28-s − 197·29-s + 74·31-s − 256·32-s + 52·34-s − 340·35-s + 227·37-s − 120·38-s − 165·41-s − 156·43-s − 256·44-s − 312·46-s − 162·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.52·5-s − 1.07·7-s + 2.15·10-s − 0.877·11-s − 1.52·14-s − 16-s + 0.185·17-s − 0.362·19-s + 1.52·20-s − 1.24·22-s − 0.707·23-s + 1.31·25-s − 1.07·28-s − 1.26·29-s + 0.428·31-s − 1.41·32-s + 0.262·34-s − 1.64·35-s + 1.00·37-s − 0.512·38-s − 0.628·41-s − 0.553·43-s − 0.877·44-s − 1.00·46-s − 0.502·47-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: yes
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 \( 1 - p^{2} T + p^{3} T^{2} \)
5 \( 1 - 17 T + p^{3} T^{2} \)
7 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 + 32 T + p^{3} T^{2} \)
17 \( 1 - 13 T + p^{3} T^{2} \)
19 \( 1 + 30 T + p^{3} T^{2} \)
23 \( 1 + 78 T + p^{3} T^{2} \)
29 \( 1 + 197 T + p^{3} T^{2} \)
31 \( 1 - 74 T + p^{3} T^{2} \)
37 \( 1 - 227 T + p^{3} T^{2} \)
41 \( 1 + 165 T + p^{3} T^{2} \)
43 \( 1 + 156 T + p^{3} T^{2} \)
47 \( 1 + 162 T + p^{3} T^{2} \)
53 \( 1 + 93 T + p^{3} T^{2} \)
59 \( 1 + 864 T + p^{3} T^{2} \)
61 \( 1 - 145 T + p^{3} T^{2} \)
67 \( 1 + 862 T + p^{3} T^{2} \)
71 \( 1 - 654 T + p^{3} T^{2} \)
73 \( 1 + 215 T + p^{3} T^{2} \)
79 \( 1 + 76 T + p^{3} T^{2} \)
83 \( 1 - 628 T + p^{3} T^{2} \)
89 \( 1 + 266 T + p^{3} T^{2} \)
97 \( 1 + 238 T + p^{3} T^{2} \)
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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

−8.920136294650283049030384821464, −7.66162194189476310668250123646, −6.49430710776968043618448412964, −6.11280905173866511994350357653, −5.44826736949723099989523146247, −4.64009483917479849855926390300, −3.47599737417239054431810970550, −2.72190868951251984021587235607, −1.87110376828255682208425364792, 0, 1.87110376828255682208425364792, 2.72190868951251984021587235607, 3.47599737417239054431810970550, 4.64009483917479849855926390300, 5.44826736949723099989523146247, 6.11280905173866511994350357653, 6.49430710776968043618448412964, 7.66162194189476310668250123646, 8.920136294650283049030384821464

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