↑

Properties

Label 2-2268-63.58-c1-0-12
Degree $2$
Conductor $2268$
Sign $0.841 - 0.540i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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 + 2.59i)7-s + (3.5 − 6.06i)13-s + (−4 + 6.92i)19-s + (2.5 − 4.33i)25-s + 11·31-s + (0.5 − 0.866i)37-s + (6.5 + 11.2i)43-s + (−6.5 − 2.59i)49-s − 61-s + 11·67-s + (5 + 8.66i)73-s − 13·79-s + (14 + 12.1i)91-s + (9.5 + 16.4i)97-s + (3.5 + 6.06i)103-s + ⋯
L(s)  = 1  + (−0.188 + 0.981i)7-s + (0.970 − 1.68i)13-s + (−0.917 + 1.58i)19-s + (0.5 − 0.866i)25-s + 1.97·31-s + (0.0821 − 0.142i)37-s + (0.991 + 1.71i)43-s + (−0.928 − 0.371i)49-s − 0.128·61-s + 1.34·67-s + (0.585 + 1.01i)73-s − 1.46·79-s + (1.46 + 1.27i)91-s + (0.964 + 1.67i)97-s + (0.344 + 0.597i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.841 - 0.540i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.841 - 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.755963055\)
\(L(\frac12)\) \(\approx\) \(1.755963055\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.5 + 6.06i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 11T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.5 - 11.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.5 - 16.4i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.946105476706226060414424613546, −8.145568567701626386864554725086, −7.993711805353816369273444181396, −6.40932566044695748795286546898, −6.08675432134947216703489736872, −5.27393562435221367877755012961, −4.21259013862628706212730304448, −3.19788529700938336785067869900, −2.41704857296097390028232057219, −1.00800530234850543059297618034, 0.77401877661380594419359150400, 2.00369312998558517200722227640, 3.22078743837634814155001143002, 4.24443792208951395271962241392, 4.65437980615657097817887047300, 5.99259733611794745766892944916, 6.84074956637456655038376477234, 7.08458385393307574404798323208, 8.339337352969809099960705614591, 8.925338174027574405539712968071

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