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Properties

Label 2-210-105.89-c1-0-7
Degree $2$
Conductor $210$
Sign $0.994 + 0.108i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (1.01 − 1.40i)3-s + (−0.499 − 0.866i)4-s + (2.20 − 0.358i)5-s + (0.707 + 1.58i)6-s + (−1.41 + 2.23i)7-s + 0.999·8-s + (−0.936 − 2.85i)9-s + (−0.792 + 2.09i)10-s + (4.05 − 2.34i)11-s + (−1.72 − 0.178i)12-s − 1.04·13-s + (−1.22 − 2.34i)14-s + (1.73 − 3.46i)15-s + (−0.5 + 0.866i)16-s + (2.73 − 1.58i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.586 − 0.809i)3-s + (−0.249 − 0.433i)4-s + (0.987 − 0.160i)5-s + (0.288 + 0.645i)6-s + (−0.534 + 0.845i)7-s + 0.353·8-s + (−0.312 − 0.950i)9-s + (−0.250 + 0.661i)10-s + (1.22 − 0.706i)11-s + (−0.497 − 0.0514i)12-s − 0.289·13-s + (−0.328 − 0.626i)14-s + (0.448 − 0.893i)15-s + (−0.125 + 0.216i)16-s + (0.664 − 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.994 + 0.108i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.994 + 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32348 - 0.0722268i\)
\(L(\frac12)\) \(\approx\) \(1.32348 - 0.0722268i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.01 + 1.40i)T \)
5 \( 1 + (-2.20 + 0.358i)T \)
7 \( 1 + (1.41 - 2.23i)T \)
good11 \( 1 + (-4.05 + 2.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.04T + 13T^{2} \)
17 \( 1 + (-2.73 + 1.58i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.23 + 0.715i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 - 3.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + (5.73 - 3.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.30 + 1.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.04T + 41T^{2} \)
43 \( 1 - 6.92iT - 43T^{2} \)
47 \( 1 + (9.71 + 5.60i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.5 + 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.28 - 9.15i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.3 - 7.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 + (1.75 + 3.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.73 - 9.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.06iT - 83T^{2} \)
89 \( 1 + (-2.45 + 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.57161064968449073041058370484, −11.57302884746521676226546317969, −9.925919831586808790113453458539, −9.111895157035453295317117776213, −8.605927342774027810495927535427, −7.15650749666247503349770139850, −6.29235593086728422889210561530, −5.46474696832450576028279739796, −3.23733276215311334751537465814, −1.59737375030954305265428154860, 1.98595525714999236684361449843, 3.50378920485052429939015174518, 4.52090362550690704382364341755, 6.20472408996847117704339010679, 7.50964006726108806255209546889, 8.830443468319713289561776562536, 9.824305407619039349832839598478, 10.02461301974698663466694312154, 11.10192870426276915253775281639, 12.41959250456675965673308771178

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