↑

Properties

Label 2-1368-1368.371-c0-0-0
Degree $2$
Conductor $1368$
Sign $-0.196 - 0.980i$
Analytic cond. $0.682720$
Root an. cond. $0.826269$
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  + (−0.939 − 0.342i)2-s − 3-s + (0.766 + 0.642i)4-s + (0.939 + 0.342i)6-s + (−0.500 − 0.866i)8-s + 9-s + 1.96i·11-s + (−0.766 − 0.642i)12-s + (0.173 + 0.984i)16-s + (−1.70 + 0.300i)17-s + (−0.939 − 0.342i)18-s + (−0.766 − 0.642i)19-s + (0.673 − 1.85i)22-s + (0.500 + 0.866i)24-s + (0.939 − 0.342i)25-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s − 3-s + (0.766 + 0.642i)4-s + (0.939 + 0.342i)6-s + (−0.500 − 0.866i)8-s + 9-s + 1.96i·11-s + (−0.766 − 0.642i)12-s + (0.173 + 0.984i)16-s + (−1.70 + 0.300i)17-s + (−0.939 − 0.342i)18-s + (−0.766 − 0.642i)19-s + (0.673 − 1.85i)22-s + (0.500 + 0.866i)24-s + (0.939 − 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.196 - 0.980i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.196 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3090701848\)
\(L(\frac12)\) \(\approx\) \(0.3090701848\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + T \)
19 \( 1 + (0.766 + 0.642i)T \)
good5 \( 1 + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 - 1.96iT - T^{2} \)
13 \( 1 + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (1.53 - 1.28i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \)
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

−10.06488635232594190340594901346, −9.382931530548520346748075603745, −8.555018078971248548471345604506, −7.44599765008771533093167597318, −6.81401629997339168899099935661, −6.29706496573432130441454895526, −4.73740580810906711519864569813, −4.28931734648859194720619653115, −2.53852787751659386216075877451, −1.57614975718305704171868553558, 0.39073901334932624721144622182, 1.88015072496478098082338367018, 3.37594740586986956057051745385, 4.78349529928446513842506712577, 5.66983882048273454402176873992, 6.43140276541744598902243038037, 6.88646138584829257336163720316, 8.122195309886226848076984508499, 8.691875640633102494333736024843, 9.492435995227763768732144858838

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