↑

Properties

Label 2-210-5.3-c2-0-3
Degree $2$
Conductor $210$
Sign $0.772 - 0.634i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (−2.20 − 4.48i)5-s − 2.44·6-s + (−1.87 + 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s + (6.69 + 2.28i)10-s + 15.1·11-s + (2.44 − 2.44i)12-s + (14.6 + 14.6i)13-s − 3.74i·14-s + (2.79 − 8.19i)15-s − 4·16-s + (15.9 − 15.9i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.440 − 0.897i)5-s − 0.408·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.669 + 0.228i)10-s + 1.37·11-s + (0.204 − 0.204i)12-s + (1.12 + 1.12i)13-s − 0.267i·14-s + (0.186 − 0.546i)15-s − 0.250·16-s + (0.940 − 0.940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.772 - 0.634i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.772 - 0.634i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.29438 + 0.463700i\)
\(L(\frac12)\) \(\approx\) \(1.29438 + 0.463700i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (2.20 + 4.48i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good11 \( 1 - 15.1T + 121T^{2} \)
13 \( 1 + (-14.6 - 14.6i)T + 169iT^{2} \)
17 \( 1 + (-15.9 + 15.9i)T - 289iT^{2} \)
19 \( 1 + 2.05iT - 361T^{2} \)
23 \( 1 + (-19.4 - 19.4i)T + 529iT^{2} \)
29 \( 1 + 33.1iT - 841T^{2} \)
31 \( 1 - 27.8T + 961T^{2} \)
37 \( 1 + (30.4 - 30.4i)T - 1.36e3iT^{2} \)
41 \( 1 + 26.6T + 1.68e3T^{2} \)
43 \( 1 + (-7.89 - 7.89i)T + 1.84e3iT^{2} \)
47 \( 1 + (-33.8 + 33.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (-5.60 - 5.60i)T + 2.80e3iT^{2} \)
59 \( 1 + 5.80iT - 3.48e3T^{2} \)
61 \( 1 + 98.2T + 3.72e3T^{2} \)
67 \( 1 + (-51.6 + 51.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 120.T + 5.04e3T^{2} \)
73 \( 1 + (-81.9 - 81.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 33.0iT - 6.24e3T^{2} \)
83 \( 1 + (97.0 + 97.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 34.2iT - 7.92e3T^{2} \)
97 \( 1 + (-104. + 104. i)T - 9.40e3iT^{2} \)
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

−11.95276997463105068185926040555, −11.45033030324746259598770959924, −9.839342055412793127791636326998, −9.106197214439302199704191040636, −8.561037829980350731979422758431, −7.30522758611349023075280055950, −6.12662398681412039814943057425, −4.76913078011863874158884449176, −3.61164725923202393317325127270, −1.29110722981586088033091311054, 1.18782990679096045559197901198, 3.09618263054485291305880923166, 3.83884350791829336078430614737, 6.15130875527573142579735626182, 7.07767013631261688548329478784, 8.140303882436260153400464077517, 8.972827996937529554496023282348, 10.31611116667279847641342614471, 10.87006115413058602387898382764, 12.04009822949052183486755338867

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