Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 $
Sign $0.854 - 0.519i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (1.93 + 1.11i)5-s + 2.44i·6-s + (6.51 + 2.55i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (−2.73 + 1.58i)10-s + (−5.79 − 10.0i)11-s + (−2.99 − 1.73i)12-s + 7.86i·13-s + (−7.73 + 6.17i)14-s + 3.87·15-s + (−2.00 + 3.46i)16-s + (23.9 − 13.8i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + 0.408i·6-s + (0.930 + 0.365i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.273 + 0.158i)10-s + (−0.526 − 0.912i)11-s + (−0.249 − 0.144i)12-s + 0.604i·13-s + (−0.552 + 0.440i)14-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (1.40 − 0.811i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.854 - 0.519i$
motivic weight  =  \(2\)
character  :  $\chi_{210} (61, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 210,\ (\ :1),\ 0.854 - 0.519i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.67002 + 0.468158i\)
\(L(\frac12)\)  \(\approx\)  \(1.67002 + 0.468158i\)
\(L(2)\)   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 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (-6.51 - 2.55i)T \)
good11 \( 1 + (5.79 + 10.0i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 7.86iT - 169T^{2} \)
17 \( 1 + (-23.9 + 13.8i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-27.2 - 15.7i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (9.07 - 15.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 2.30T + 841T^{2} \)
31 \( 1 + (4.55 - 2.63i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (0.993 - 1.72i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 22.1iT - 1.68e3T^{2} \)
43 \( 1 - 49.8T + 1.84e3T^{2} \)
47 \( 1 + (66.3 + 38.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (28.5 + 49.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (60.9 - 35.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (58.5 + 33.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (49.0 + 85.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 34.2T + 5.04e3T^{2} \)
73 \( 1 + (-16.8 + 9.74i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (45.2 - 78.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 133. iT - 6.88e3T^{2} \)
89 \( 1 + (9.58 + 5.53i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 72.3iT - 9.40e3T^{2} \)
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

−12.14543969009725817167349434082, −11.27195430379320999167481075859, −9.982245015109566649697775805908, −9.171137671977348005425333489703, −8.005341581470028477006457623198, −7.50001336827152904703243624613, −5.98532046867157130867750495682, −5.13543427818680149728389361528, −3.21461877769139074753383182254, −1.47281166997381092443151912662, 1.41282984720215686753738683983, 2.89801538916378466278269402778, 4.39680578299410982993962307567, 5.45294358066515848147870183319, 7.48359706445293905166584920309, 8.061305893068659487698981973175, 9.290390899216182549301719602233, 10.13063421627308673157796324676, 10.83108297589851661710885051778, 12.07739147056559738270753298320

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