↑

Properties

Label 2-210-21.20-c1-0-0
Degree $2$
Conductor $210$
Sign $0.218 - 0.975i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
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  − i·2-s + (−0.618 + 1.61i)3-s − 4-s − 5-s + (1.61 + 0.618i)6-s + (0.381 + 2.61i)7-s + i·8-s + (−2.23 − 2.00i)9-s + i·10-s + 4.47i·11-s + (0.618 − 1.61i)12-s + 3.23i·13-s + (2.61 − 0.381i)14-s + (0.618 − 1.61i)15-s + 16-s − 0.763·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.356 + 0.934i)3-s − 0.5·4-s − 0.447·5-s + (0.660 + 0.252i)6-s + (0.144 + 0.989i)7-s + 0.353i·8-s + (−0.745 − 0.666i)9-s + 0.316i·10-s + 1.34i·11-s + (0.178 − 0.467i)12-s + 0.897i·13-s + (0.699 − 0.102i)14-s + (0.159 − 0.417i)15-s + 0.250·16-s − 0.185·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.218 - 0.975i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.218 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629388 + 0.504195i\)
\(L(\frac12)\) \(\approx\) \(0.629388 + 0.504195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.618 - 1.61i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.381 - 2.61i)T \)
good11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 - 3.23iT - 13T^{2} \)
17 \( 1 + 0.763T + 17T^{2} \)
19 \( 1 + 0.472iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 5.70iT - 29T^{2} \)
31 \( 1 + 7.23iT - 31T^{2} \)
37 \( 1 - 5.23T + 37T^{2} \)
41 \( 1 + 6.47T + 41T^{2} \)
43 \( 1 - 12.9T + 43T^{2} \)
47 \( 1 + 2.47T + 47T^{2} \)
53 \( 1 - 8.47iT - 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 - 2.76iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 2.76iT - 71T^{2} \)
73 \( 1 + 6.76iT - 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 - 5.23iT - 97T^{2} \)
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

−12.01840924437992871122136638973, −11.79032693231415015685817541672, −10.74325855418097507554286972674, −9.547487619612675640524745244220, −9.164419542776622853630167849868, −7.74747172114191661064883955424, −6.11185740583984162563712274414, −4.83596773642701562228480550864, −4.01044565414653378996205278864, −2.36425259544367887198175546683, 0.74380502191312902225374957915, 3.33634467327170917169484353562, 4.96317959414577807808508999949, 6.13616947239813470342391372262, 7.08277001978161789814923756476, 7.985756064625003225791222056715, 8.688746073863802366844478885316, 10.49285607584865088829789938026, 11.12354476211439981238924812117, 12.37981067120107325600304814535

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