↑

Properties

Label 2-2268-252.167-c0-0-15
Degree $2$
Conductor $2268$
Sign $-0.573 + 0.819i$
Analytic cond. $1.13187$
Root an. cond. $1.06389$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (−0.965 − 1.67i)11-s + (0.258 − 0.965i)14-s − 1.00·16-s + (−1.86 − 0.5i)22-s + (−0.707 + 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.500 − 0.866i)28-s + (1.22 − 0.707i)29-s + (−0.707 + 0.707i)32-s − 1.73·37-s + (0.866 − 0.5i)43-s + (−1.67 + 0.965i)44-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (−0.965 − 1.67i)11-s + (0.258 − 0.965i)14-s − 1.00·16-s + (−1.86 − 0.5i)22-s + (−0.707 + 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.500 − 0.866i)28-s + (1.22 − 0.707i)29-s + (−0.707 + 0.707i)32-s − 1.73·37-s + (0.866 − 0.5i)43-s + (−1.67 + 0.965i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.573 + 0.819i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ -0.573 + 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.667104263\)
\(L(\frac12)\) \(\approx\) \(1.667104263\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.73T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 0.517iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 0.517T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−8.929617924683150521237223569194, −8.248931850162411785548581880374, −7.42310595935724917594437838537, −6.37815165664857791779472626114, −5.45694817495751125652735606910, −5.07079356062829701877578410720, −3.89157385439772137653139084149, −3.24734083937273219926686958261, −2.14712113484770154133508489803, −0.926874178966851079306921257376, 2.06714648943325305593715585546, 2.75277650022105899339213117446, 4.17658354756482092149860697835, 4.83557589754510976295479160592, 5.29987222709464301476655681942, 6.44052338948112642105036865791, 7.05217365338238017793656600890, 7.974638944874001306677050356979, 8.345555361899976952425562350127, 9.268353886437991834276083914230

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