Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Sign $-0.894 + 0.447i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + i·3-s − 4-s + (2 − i)5-s − 6-s i·8-s − 9-s + (1 + 2i)10-s − 5·11-s i·12-s + i·13-s + (1 + 2i)15-s + 16-s + 2i·17-s i·18-s − 7·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.894 − 0.447i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.316 + 0.632i)10-s − 1.50·11-s − 0.288i·12-s + 0.277i·13-s + (0.258 + 0.516i)15-s + 0.250·16-s + 0.485i·17-s − 0.235i·18-s − 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.894 + 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{1470} (589, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1470,\ (\ :1/2),\ -0.894 + 0.447i)\)
\(L(1)\)  \(\approx\)  \(0.6126547694\)
\(L(\frac12)\)  \(\approx\)  \(0.6126547694\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 + (-2 + i)T \)
7 \( 1 \)
good11 \( 1 + 5T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 5iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 - 13iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.886854455635435559973941916322, −9.105756100425484653667666449819, −8.388886862842258079571312699229, −7.70976505900251366710457066509, −6.48564632037147651937050486620, −5.86435158814388079943570036739, −5.02093583675257517395915830129, −4.42043626204439815208479484311, −3.07332614542698077553300296824, −1.83193334314084235248106097083, 0.21987276389889351163085446506, 2.00821111096943333565913103016, 2.47132488234768424377477649349, 3.59052577848097768589292253766, 5.00908477864696606905056858633, 5.60923192592448998276820419979, 6.61853468352170945919737778273, 7.42151430090843061699936346803, 8.411092767280935435889115081642, 9.027115059790585158195686307713

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