Properties

Degree 2
Conductor $ 3 \cdot 7 $
Sign $0.607 - 0.794i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.30i·2-s + (−1.82 + 2.38i)3-s + 2.29·4-s − 7.37i·5-s + (−3.11 − 2.38i)6-s − 2.64·7-s + 8.22i·8-s + (−2.35 − 8.68i)9-s + 9.64·10-s + 2.61i·11-s + (−4.17 + 5.45i)12-s − 6.35·13-s − 3.45i·14-s + (17.5 + 13.4i)15-s − 1.58·16-s + 12.1i·17-s + ⋯
L(s)  = 1  + 0.653i·2-s + (−0.607 + 0.794i)3-s + 0.572·4-s − 1.47i·5-s + (−0.519 − 0.397i)6-s − 0.377·7-s + 1.02i·8-s + (−0.261 − 0.965i)9-s + 0.964·10-s + 0.237i·11-s + (−0.348 + 0.454i)12-s − 0.488·13-s − 0.247i·14-s + (1.17 + 0.896i)15-s − 0.0989·16-s + 0.714i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $0.607 - 0.794i$
motivic weight  =  \(2\)
character  :  $\chi_{21} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 21,\ (\ :1),\ 0.607 - 0.794i)$
$L(\frac{3}{2})$  $\approx$  $0.742454 + 0.366798i$
$L(\frac12)$  $\approx$  $0.742454 + 0.366798i$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (1.82 - 2.38i)T \)
7 \( 1 + 2.64T \)
good2 \( 1 - 1.30iT - 4T^{2} \)
5 \( 1 + 7.37iT - 25T^{2} \)
11 \( 1 - 2.61iT - 121T^{2} \)
13 \( 1 + 6.35T + 169T^{2} \)
17 \( 1 - 12.1iT - 289T^{2} \)
19 \( 1 + 10.2T + 361T^{2} \)
23 \( 1 + 4.30iT - 529T^{2} \)
29 \( 1 + 17.3iT - 841T^{2} \)
31 \( 1 - 39.2T + 961T^{2} \)
37 \( 1 - 41.0T + 1.36e3T^{2} \)
41 \( 1 - 30.2iT - 1.68e3T^{2} \)
43 \( 1 + 55.8T + 1.84e3T^{2} \)
47 \( 1 - 39.9iT - 2.20e3T^{2} \)
53 \( 1 + 105. iT - 2.80e3T^{2} \)
59 \( 1 + 41.3iT - 3.48e3T^{2} \)
61 \( 1 + 20.4T + 3.72e3T^{2} \)
67 \( 1 + 27.1T + 4.48e3T^{2} \)
71 \( 1 - 67.8iT - 5.04e3T^{2} \)
73 \( 1 - 60.7T + 5.32e3T^{2} \)
79 \( 1 + 63.2T + 6.24e3T^{2} \)
83 \( 1 + 89.9iT - 6.88e3T^{2} \)
89 \( 1 + 63.1iT - 7.92e3T^{2} \)
97 \( 1 - 19.1T + 9.40e3T^{2} \)
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

−17.39330515913206427519390381782, −16.75110468915674496519824156456, −15.90918020796956462301426052915, −14.87537043484530243946296448027, −12.77676763769673097375325456987, −11.61055449222500752374842862982, −9.912622864366688966299283425911, −8.365753564601864572636346836896, −6.23009728629170269715395110239, −4.76189951599963393312237247570, 2.72394693730293101925304774111, 6.35514300323772687842844043143, 7.31850106587427905497581587199, 10.22044971433123207637602782910, 11.18926705230815417960428913541, 12.21212455621933931337508124739, 13.70199429714925516254534406272, 15.23636172881219556133467040338, 16.69668689372146167554062418988, 18.21700636530664691977716595000

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