Properties

Label 2-285-1.1-c1-0-8
Degree $2$
Conductor $285$
Sign $-1$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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 − 2·7-s + 3·8-s + 9-s + 10-s − 6·11-s − 12-s + 2·14-s − 15-s − 16-s − 6·17-s − 18-s + 19-s + 20-s − 2·21-s + 6·22-s − 8·23-s + 3·24-s + 25-s + 27-s + 2·28-s + 4·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.80·11-s − 0.288·12-s + 0.534·14-s − 0.258·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.436·21-s + 1.27·22-s − 1.66·23-s + 0.612·24-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.742·29-s + 0.182·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{285} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−11.04359797233522452957058795898, −10.14916660629762825787921129871, −9.508739877617597277399182494308, −8.311172749518175221259006108214, −7.944956574862892187546933534941, −6.69343741358824607018216997419, −5.08269165937541766478904797647, −3.93065235513561826916389859535, −2.45875404104195454497297562929, 0, 2.45875404104195454497297562929, 3.93065235513561826916389859535, 5.08269165937541766478904797647, 6.69343741358824607018216997419, 7.944956574862892187546933534941, 8.311172749518175221259006108214, 9.508739877617597277399182494308, 10.14916660629762825787921129871, 11.04359797233522452957058795898

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