Properties

Label 2-285-1.1-c1-0-9
Degree $2$
Conductor $285$
Sign $1$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·2-s + 3-s + 3.82·4-s − 5-s + 2.41·6-s − 1.41·7-s + 4.41·8-s + 9-s − 2.41·10-s − 2.24·11-s + 3.82·12-s − 3.41·13-s − 3.41·14-s − 15-s + 2.99·16-s + 1.17·17-s + 2.41·18-s − 19-s − 3.82·20-s − 1.41·21-s − 5.41·22-s + 7.65·23-s + 4.41·24-s + 25-s − 8.24·26-s + 27-s − 5.41·28-s + ⋯
L(s)  = 1  + 1.70·2-s + 0.577·3-s + 1.91·4-s − 0.447·5-s + 0.985·6-s − 0.534·7-s + 1.56·8-s + 0.333·9-s − 0.763·10-s − 0.676·11-s + 1.10·12-s − 0.946·13-s − 0.912·14-s − 0.258·15-s + 0.749·16-s + 0.284·17-s + 0.569·18-s − 0.229·19-s − 0.856·20-s − 0.308·21-s − 1.15·22-s + 1.59·23-s + 0.901·24-s + 0.200·25-s − 1.61·26-s + 0.192·27-s − 1.02·28-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: no
Arithmetic: yes
Character: $\chi_{285} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.216419366\)
\(L(\frac12)\) \(\approx\) \(3.216419366\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 2T^{2} \)
7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
23 \( 1 - 7.65T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 3.17T + 31T^{2} \)
37 \( 1 + 3.41T + 37T^{2} \)
41 \( 1 + 0.242T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + 7.65T + 47T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 - 7.31T + 61T^{2} \)
67 \( 1 + 9.65T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 7.65T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 - 5.89T + 89T^{2} \)
97 \( 1 + 9.75T + 97T^{2} \)
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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

−12.23511320145276539552922414293, −11.21411063461569254226128164985, −10.19067810920104136365419417778, −8.927516763719285408940852117938, −7.54003299218301302356581633802, −6.83378106786382026747386148724, −5.49238882760750975616135169751, −4.56926424233502251010085086314, −3.40287857323995405053371516534, −2.53627796200111631637271098706, 2.53627796200111631637271098706, 3.40287857323995405053371516534, 4.56926424233502251010085086314, 5.49238882760750975616135169751, 6.83378106786382026747386148724, 7.54003299218301302356581633802, 8.927516763719285408940852117938, 10.19067810920104136365419417778, 11.21411063461569254226128164985, 12.23511320145276539552922414293

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