Properties

Label 2-39e2-1.1-c3-0-103
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

−8.597106306167074020962821009361, −7.998956244654159954918217383937, −7.47053540456839375377141010333, −6.78971942676496127619296019830, −5.35118800259356119955475955179, −4.29773905754234397348738244275, −3.63462630310652193132796683210, −2.00307579042291979297885844202, −1.04252772225174963743583336470, 0, 1.04252772225174963743583336470, 2.00307579042291979297885844202, 3.63462630310652193132796683210, 4.29773905754234397348738244275, 5.35118800259356119955475955179, 6.78971942676496127619296019830, 7.47053540456839375377141010333, 7.998956244654159954918217383937, 8.597106306167074020962821009361

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