Properties

Label 2-285-1.1-c1-0-3
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s + 3-s − 1.82·4-s − 5-s − 0.414·6-s + 1.41·7-s + 1.58·8-s + 9-s + 0.414·10-s + 6.24·11-s − 1.82·12-s − 0.585·13-s − 0.585·14-s − 15-s + 3·16-s + 6.82·17-s − 0.414·18-s − 19-s + 1.82·20-s + 1.41·21-s − 2.58·22-s − 3.65·23-s + 1.58·24-s + 25-s + 0.242·26-s + 27-s − 2.58·28-s + ⋯
L(s)  = 1  − 0.292·2-s + 0.577·3-s − 0.914·4-s − 0.447·5-s − 0.169·6-s + 0.534·7-s + 0.560·8-s + 0.333·9-s + 0.130·10-s + 1.88·11-s − 0.527·12-s − 0.162·13-s − 0.156·14-s − 0.258·15-s + 0.750·16-s + 1.65·17-s − 0.0976·18-s − 0.229·19-s + 0.408·20-s + 0.308·21-s − 0.551·22-s − 0.762·23-s + 0.323·24-s + 0.200·25-s + 0.0475·26-s + 0.192·27-s − 0.488·28-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

\(L(1)\) \(\approx\) \(1.192554221\)
\(L(\frac12)\) \(\approx\) \(1.192554221\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 2T^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 - 6.24T + 11T^{2} \)
13 \( 1 + 0.585T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 + 1.41T + 29T^{2} \)
31 \( 1 + 8.82T + 31T^{2} \)
37 \( 1 + 0.585T + 37T^{2} \)
41 \( 1 - 8.24T + 41T^{2} \)
43 \( 1 - 3.75T + 43T^{2} \)
47 \( 1 - 3.65T + 47T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + 4.48T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 - 1.65T + 67T^{2} \)
71 \( 1 + 5.17T + 71T^{2} \)
73 \( 1 - 3.65T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 7.17T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 18.2T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.97617666403385932581912751005, −10.78753297735059819187015212485, −9.600468563980552849636352847966, −9.061853962899491974237723894976, −8.071291712576376438194928429427, −7.32162446601290164924871155949, −5.74500436770186020896138892320, −4.32766183425900849295621201115, −3.60433233639417382193935484750, −1.38227551095877088266721583956, 1.38227551095877088266721583956, 3.60433233639417382193935484750, 4.32766183425900849295621201115, 5.74500436770186020896138892320, 7.32162446601290164924871155949, 8.071291712576376438194928429427, 9.061853962899491974237723894976, 9.600468563980552849636352847966, 10.78753297735059819187015212485, 11.97617666403385932581912751005

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