↑

Properties

Label 2-1368-152.75-c1-0-23
Degree $2$
Conductor $1368$
Sign $-i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
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  + 1.41i·2-s − 2.00·4-s − 1.13i·5-s − 2.82i·8-s + 1.60·10-s − 5.89·13-s + 4.00·16-s + 4.35·19-s + 2.26i·20-s + 9.46i·23-s + 3.71·25-s − 8.33i·26-s + 4.29·31-s + 5.65i·32-s + 9.09·37-s + 6.16i·38-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s − 0.506i·5-s − 1.00i·8-s + 0.506·10-s − 1.63·13-s + 1.00·16-s + 1.00·19-s + 0.506i·20-s + 1.97i·23-s + 0.743·25-s − 1.63i·26-s + 0.770·31-s + 1.00i·32-s + 1.49·37-s + 1.00i·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.306581774\)
\(L(\frac12)\) \(\approx\) \(1.306581774\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
19 \( 1 - 4.35T \)
good5 \( 1 + 1.13iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 5.89T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 - 9.46iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4.29T + 31T^{2} \)
37 \( 1 - 9.09T + 37T^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 - 8.71T + 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 2.82iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 8.71T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 11.3iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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

−9.571588144247104078615998724825, −9.035952108296814250316173803950, −7.86896158159437545767048168298, −7.52222359108773772115228331232, −6.60117766361373818961389438860, −5.49725013325709232699795986171, −5.01329121434853521030263868952, −4.07556818700369521647231195304, −2.81748546937369728370324110516, −1.01933378946581885926344892795, 0.69804711655637323897437978396, 2.42011938644725166932866359784, 2.85912795875186941622982756667, 4.22662267552091816160883959094, 4.88687875252548583550734727158, 5.93562448345851539909348509655, 7.09523890921209785301832276867, 7.82038476809766438227133556805, 8.843750476598122458783783617555, 9.564054366066768880738604818300

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