Properties

Label 2-285-1.1-c3-0-8
Degree $2$
Conductor $285$
Sign $1$
Analytic cond. $16.8155$
Root an. cond. $4.10067$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4-s + 5·5-s + 9·6-s + 32·7-s + 21·8-s + 9·9-s − 15·10-s − 12·11-s − 3·12-s − 10·13-s − 96·14-s − 15·15-s − 71·16-s − 30·17-s − 27·18-s + 19·19-s + 5·20-s − 96·21-s + 36·22-s − 48·23-s − 63·24-s + 25·25-s + 30·26-s − 27·27-s + 32·28-s + ⋯
L(s)  = 1  − 1.06·2-s − 0.577·3-s + 1/8·4-s + 0.447·5-s + 0.612·6-s + 1.72·7-s + 0.928·8-s + 1/3·9-s − 0.474·10-s − 0.328·11-s − 0.0721·12-s − 0.213·13-s − 1.83·14-s − 0.258·15-s − 1.10·16-s − 0.428·17-s − 0.353·18-s + 0.229·19-s + 0.0559·20-s − 0.997·21-s + 0.348·22-s − 0.435·23-s − 0.535·24-s + 1/5·25-s + 0.226·26-s − 0.192·27-s + 0.215·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(16.8155\)
Root analytic conductor: \(4.10067\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.012139777\)
\(L(\frac12)\) \(\approx\) \(1.012139777\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p T \)
5 \( 1 - p T \)
19 \( 1 - p T \)
good2 \( 1 + 3 T + p^{3} T^{2} \)
7 \( 1 - 32 T + p^{3} T^{2} \)
11 \( 1 + 12 T + p^{3} T^{2} \)
13 \( 1 + 10 T + p^{3} T^{2} \)
17 \( 1 + 30 T + p^{3} T^{2} \)
23 \( 1 + 48 T + p^{3} T^{2} \)
29 \( 1 - 150 T + p^{3} T^{2} \)
31 \( 1 - 224 T + p^{3} T^{2} \)
37 \( 1 - 254 T + p^{3} T^{2} \)
41 \( 1 + 54 T + p^{3} T^{2} \)
43 \( 1 + 196 T + p^{3} T^{2} \)
47 \( 1 + 504 T + p^{3} T^{2} \)
53 \( 1 - 78 T + p^{3} T^{2} \)
59 \( 1 - 132 T + p^{3} T^{2} \)
61 \( 1 - 230 T + p^{3} T^{2} \)
67 \( 1 - 740 T + p^{3} T^{2} \)
71 \( 1 + 120 T + p^{3} T^{2} \)
73 \( 1 - 122 T + p^{3} T^{2} \)
79 \( 1 - 1184 T + p^{3} T^{2} \)
83 \( 1 - 612 T + p^{3} T^{2} \)
89 \( 1 - 1050 T + p^{3} T^{2} \)
97 \( 1 + 1006 T + p^{3} T^{2} \)
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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

−11.17351005151422656070833001596, −10.39060470497512782057109782028, −9.593586232281630386263716738709, −8.362589237597834458519371717003, −7.902528036024335588534948171376, −6.63654967159124734404929020933, −5.16706771820010994153897214217, −4.51547998721895948351379796713, −2.07125330776242329486876369380, −0.905160415962657082661860665831, 0.905160415962657082661860665831, 2.07125330776242329486876369380, 4.51547998721895948351379796713, 5.16706771820010994153897214217, 6.63654967159124734404929020933, 7.902528036024335588534948171376, 8.362589237597834458519371717003, 9.593586232281630386263716738709, 10.39060470497512782057109782028, 11.17351005151422656070833001596

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