↑

Properties

Label 2-435-145.144-c1-0-8
Degree $2$
Conductor $435$
Sign $0.0830 - 0.996i$
Analytic cond. $3.47349$
Root an. cond. $1.86373$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s + (2 + i)5-s − 2·6-s + 4i·7-s + 9-s + (−4 − 2i)10-s + i·11-s + 2·12-s + 2i·13-s − 8i·14-s + (2 + i)15-s − 4·16-s − 6·17-s − 2·18-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s + (0.894 + 0.447i)5-s − 0.816·6-s + 1.51i·7-s + 0.333·9-s + (−1.26 − 0.632i)10-s + 0.301i·11-s + 0.577·12-s + 0.554i·13-s − 2.13i·14-s + (0.516 + 0.258i)15-s − 16-s − 1.45·17-s − 0.471·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0830 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0830 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.0830 - 0.996i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.0830 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.645544 + 0.593986i\)
\(L(\frac12)\) \(\approx\) \(0.645544 + 0.593986i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
29 \( 1 + (2 + 5i)T \)
good2 \( 1 + 2T + 2T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 9iT - 23T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 9iT - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 15T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + 7iT - 83T^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 - 11T + 97T^{2} \)
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

−10.97354914286742907840547048921, −10.16611440680651221428335497561, −9.307911693400769796865292465243, −8.875556592568695344666217360988, −8.095690002938542576854341536020, −6.82866517287244109916693696737, −6.09095042061882719816334988417, −4.57114576488874229321384665906, −2.52509541441722889404633038325, −1.96327910702602339931425882472, 0.834978617783562150213868522314, 2.11783780223542915033328418052, 3.85144277930253994974000303287, 5.15598333046146503472668996480, 6.79568999643739799447282039592, 7.38245951446731992835070226902, 8.435083951764939988995447218328, 9.158389456161346471318345142439, 9.825559453686683173643219770104, 10.64376853280538732873835196580

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