↑

Properties

Label 2-2106-9.4-c1-0-8
Degree $2$
Conductor $2106$
Sign $-0.173 - 0.984i$
Analytic cond. $16.8164$
Root an. cond. $4.10079$
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 + (−0.499 + 0.866i)4-s + (−1.5 + 2.59i)5-s + (0.5 + 0.866i)7-s + 0.999·8-s + 3·10-s + (3 + 5.19i)11-s + (−0.5 + 0.866i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 3·17-s + 2·19-s + (−1.50 − 2.59i)20-s + (3 − 5.19i)22-s + (−2 − 3.46i)25-s + 0.999·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.670 + 1.16i)5-s + (0.188 + 0.327i)7-s + 0.353·8-s + 0.948·10-s + (0.904 + 1.56i)11-s + (−0.138 + 0.240i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.727·17-s + 0.458·19-s + (−0.335 − 0.580i)20-s + (0.639 − 1.10i)22-s + (−0.400 − 0.692i)25-s + 0.196·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2106\)    =    \(2 \cdot 3^{4} \cdot 13\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(16.8164\)
Root analytic conductor: \(4.10079\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2106} (1405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2106,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.159781161\)
\(L(\frac12)\) \(\approx\) \(1.159781161\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.462279417180747473478856303911, −8.652629167678642986276071030485, −7.62549428258090011742442878394, −7.17850945404415015041854043292, −6.46163243316628601439587280858, −5.14673698695617610133305378831, −4.20722828422650329334961043772, −3.42475998419603825375071362981, −2.50994507194685297682424288031, −1.45256340996681945797555203739, 0.54067676372901253582954523560, 1.29900916600629197374557757685, 3.22921395758859085335288971148, 4.08880389106126506220649867300, 4.95968628888575496035496313293, 5.73363208443758482826788879221, 6.53117745871754265833087060861, 7.58037448037477428082984804782, 8.155889650143300288587153554201, 8.750522579260524808751623276279

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