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Properties

Label 2-20535-1.1-c1-0-4
Degree $2$
Conductor $20535$
Sign $-1$
Analytic cond. $163.972$
Root an. cond. $12.8051$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 5-s − 6-s − 3·8-s + 9-s − 10-s − 4·11-s + 12-s + 2·13-s + 15-s − 16-s − 2·17-s + 18-s − 4·19-s + 20-s − 4·22-s + 3·24-s + 25-s + 2·26-s − 27-s + 2·29-s + 30-s + 5·32-s + 4·33-s − 2·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.852·22-s + 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.182·30-s + 0.883·32-s + 0.696·33-s − 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20535\)    =    \(3 \cdot 5 \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(163.972\)
Root analytic conductor: \(12.8051\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 20535,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
37 \( 1 \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.75286343800644, −15.49358559518228, −14.70459376458358, −14.33704472658271, −13.52302886694900, −13.14813686323856, −12.70814591199796, −12.24837160169599, −11.58016162275361, −10.88691868852222, −10.66850797768878, −9.809326770708670, −9.241221693608376, −8.489832592785392, −8.089424718618664, −7.395636635152581, −6.467813655465241, −6.163063777950260, −5.327323804538144, −4.907311057230113, −4.266480783792434, −3.719585262536123, −2.899870775467744, −2.144658226991613, −0.8192973834585261, 0, 0.8192973834585261, 2.144658226991613, 2.899870775467744, 3.719585262536123, 4.266480783792434, 4.907311057230113, 5.327323804538144, 6.163063777950260, 6.467813655465241, 7.395636635152581, 8.089424718618664, 8.489832592785392, 9.241221693608376, 9.809326770708670, 10.66850797768878, 10.88691868852222, 11.58016162275361, 12.24837160169599, 12.70814591199796, 13.14813686323856, 13.52302886694900, 14.33704472658271, 14.70459376458358, 15.49358559518228, 15.75286343800644

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