↑

Properties

Label 2-210-105.59-c1-0-11
Degree $2$
Conductor $210$
Sign $0.843 + 0.537i$
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  + (0.5 + 0.866i)2-s + (0.5 − 1.65i)3-s + (−0.499 + 0.866i)4-s + (−0.686 − 2.12i)5-s + (1.68 − 0.396i)6-s + (2.5 − 0.866i)7-s − 0.999·8-s + (−2.5 − 1.65i)9-s + (1.5 − 1.65i)10-s + (−0.813 − 0.469i)11-s + (1.18 + 1.26i)12-s + 2·13-s + (2 + 1.73i)14-s + (−3.87 + 0.0737i)15-s + (−0.5 − 0.866i)16-s + (5.74 + 3.31i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.957i)3-s + (−0.249 + 0.433i)4-s + (−0.306 − 0.951i)5-s + (0.688 − 0.161i)6-s + (0.944 − 0.327i)7-s − 0.353·8-s + (−0.833 − 0.552i)9-s + (0.474 − 0.524i)10-s + (−0.245 − 0.141i)11-s + (0.342 + 0.364i)12-s + 0.554·13-s + (0.534 + 0.462i)14-s + (−0.999 + 0.0190i)15-s + (−0.125 − 0.216i)16-s + (1.39 + 0.804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.843 + 0.537i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.843 + 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45691 - 0.425206i\)
\(L(\frac12)\) \(\approx\) \(1.45691 - 0.425206i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 1.65i)T \)
5 \( 1 + (0.686 + 2.12i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
good11 \( 1 + (0.813 + 0.469i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-5.74 - 3.31i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.686 + 1.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.31iT - 29T^{2} \)
31 \( 1 + (-6.55 - 3.78i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.11 - 4.10i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.37T + 41T^{2} \)
43 \( 1 - 1.08iT - 43T^{2} \)
47 \( 1 + (-7.37 + 4.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.18 - 3.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.55 + 11.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.0 - 6.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.05 - 1.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.51iT - 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.55 + 7.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (-0.686 - 1.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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

−12.37813835629914082468954801851, −11.80835264626607245782309133980, −10.38904240172216159030936714452, −8.573822464289021449985997612454, −8.331434835036883978846217261003, −7.35946687067289968915583909942, −6.05733348575990498407424945560, −4.99777065149639036029178311490, −3.60954262628112269371643397405, −1.41862689644502559262185222382, 2.45851440435745749479068130137, 3.61659989837218548998120229511, 4.77850208837961334392755768453, 5.89491105206299944817048796683, 7.61027248529782336179099899338, 8.618265886824722937463030351511, 9.847922590478358104195622816900, 10.60568894711880572005911133413, 11.39325845210711773296422366108, 12.08837441307158695939628376729

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