↑

Properties

Label 2-945-105.44-c0-0-3
Degree $2$
Conductor $945$
Sign $-0.660 + 0.750i$
Analytic cond. $0.471616$
Root an. cond. $0.686743$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (−0.965 + 0.258i)5-s − i·7-s + (−0.366 + 1.36i)10-s − i·13-s + (−1.22 − 0.707i)14-s + (0.499 − 0.866i)16-s + (−0.707 − 1.22i)17-s + (0.707 + 0.707i)20-s + (0.866 − 0.499i)25-s + (−1.22 − 0.707i)26-s + (−0.866 + 0.499i)28-s + 1.41i·29-s + (0.5 + 0.866i)31-s + (−0.707 − 1.22i)32-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (−0.965 + 0.258i)5-s − i·7-s + (−0.366 + 1.36i)10-s − i·13-s + (−1.22 − 0.707i)14-s + (0.499 − 0.866i)16-s + (−0.707 − 1.22i)17-s + (0.707 + 0.707i)20-s + (0.866 − 0.499i)25-s + (−1.22 − 0.707i)26-s + (−0.866 + 0.499i)28-s + 1.41i·29-s + (0.5 + 0.866i)31-s + (−0.707 − 1.22i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.660 + 0.750i$
Analytic conductor: \(0.471616\)
Root analytic conductor: \(0.686743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :0),\ -0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.233346779\)
\(L(\frac12)\) \(\approx\) \(1.233346779\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + iT \)
good2 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - iT - T^{2} \)
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

−10.40992765833599898259609264822, −9.438785241409092119662161882844, −8.150843691133203713767395856717, −7.45480288850800164427673310506, −6.61763682452520045900206714826, −4.98680938332582365394871300663, −4.47622048953190884563003704876, −3.38481002041225447329156167167, −2.83767366652793367590392593449, −1.04732421159582330389029566494, 2.16347566830592833418594702396, 3.90472329811664455685918037025, 4.38773093338556766210026176799, 5.49480637324583059877835317335, 6.23620476636196969847436548663, 7.02839980733303439784427057113, 7.986695221952279478548030996542, 8.531685895417056732620443232688, 9.381289482233697668722366996649, 10.69483117177036742121311902740

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