Properties

Label 2-285-1.1-c5-0-52
Degree $2$
Conductor $285$
Sign $-1$
Analytic cond. $45.7093$
Root an. cond. $6.76087$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 9·3-s − 28·4-s + 25·5-s + 18·6-s + 22·7-s − 120·8-s + 81·9-s + 50·10-s − 568·11-s − 252·12-s + 1.04e3·13-s + 44·14-s + 225·15-s + 656·16-s − 428·17-s + 162·18-s − 361·19-s − 700·20-s + 198·21-s − 1.13e3·22-s − 4.24e3·23-s − 1.08e3·24-s + 625·25-s + 2.09e3·26-s + 729·27-s − 616·28-s + ⋯
L(s)  = 1  + 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.447·5-s + 0.204·6-s + 0.169·7-s − 0.662·8-s + 1/3·9-s + 0.158·10-s − 1.41·11-s − 0.505·12-s + 1.71·13-s + 0.0599·14-s + 0.258·15-s + 0.640·16-s − 0.359·17-s + 0.117·18-s − 0.229·19-s − 0.391·20-s + 0.0979·21-s − 0.500·22-s − 1.67·23-s − 0.382·24-s + 1/5·25-s + 0.606·26-s + 0.192·27-s − 0.148·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(45.7093\)
Root analytic conductor: \(6.76087\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 285,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p^{2} T \)
5 \( 1 - p^{2} T \)
19 \( 1 + p^{2} T \)
good2 \( 1 - p T + p^{5} T^{2} \)
7 \( 1 - 22 T + p^{5} T^{2} \)
11 \( 1 + 568 T + p^{5} T^{2} \)
13 \( 1 - 1046 T + p^{5} T^{2} \)
17 \( 1 + 428 T + p^{5} T^{2} \)
23 \( 1 + 4244 T + p^{5} T^{2} \)
29 \( 1 + 5110 T + p^{5} T^{2} \)
31 \( 1 + 378 T + p^{5} T^{2} \)
37 \( 1 - 8622 T + p^{5} T^{2} \)
41 \( 1 + 5448 T + p^{5} T^{2} \)
43 \( 1 + 18724 T + p^{5} T^{2} \)
47 \( 1 + 13428 T + p^{5} T^{2} \)
53 \( 1 + 16774 T + p^{5} T^{2} \)
59 \( 1 - 11030 T + p^{5} T^{2} \)
61 \( 1 - 10682 T + p^{5} T^{2} \)
67 \( 1 + 54268 T + p^{5} T^{2} \)
71 \( 1 + 3528 T + p^{5} T^{2} \)
73 \( 1 - 75886 T + p^{5} T^{2} \)
79 \( 1 + 89450 T + p^{5} T^{2} \)
83 \( 1 + 57294 T + p^{5} T^{2} \)
89 \( 1 - 18540 T + p^{5} T^{2} \)
97 \( 1 + 52798 T + p^{5} T^{2} \)
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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

−10.35158413535867663387278967108, −9.545726395178645332655436350589, −8.454118781173778604807929254304, −7.987997949091572316846776140082, −6.28634743186994836748454072332, −5.38135853012022335423478613700, −4.21459254692719260512033182358, −3.18307755789966909750985457104, −1.73854863111500632776682933224, 0, 1.73854863111500632776682933224, 3.18307755789966909750985457104, 4.21459254692719260512033182358, 5.38135853012022335423478613700, 6.28634743186994836748454072332, 7.987997949091572316846776140082, 8.454118781173778604807929254304, 9.545726395178645332655436350589, 10.35158413535867663387278967108

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