↑

Properties

Label 2-1368-8.5-c1-0-41
Degree $2$
Conductor $1368$
Sign $-0.552 - 0.833i$
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  + (−0.456 + 1.33i)2-s + (−1.58 − 1.22i)4-s + 3.36i·5-s + 4.47·7-s + (2.35 − 1.56i)8-s + (−4.50 − 1.53i)10-s + 0.608i·11-s + 1.03i·13-s + (−2.04 + 5.98i)14-s + (1.01 + 3.86i)16-s + 3.06·17-s + i·19-s + (4.11 − 5.33i)20-s + (−0.814 − 0.277i)22-s + 8.50·23-s + ⋯
L(s)  = 1  + (−0.322 + 0.946i)2-s + (−0.791 − 0.610i)4-s + 1.50i·5-s + 1.68·7-s + (0.833 − 0.552i)8-s + (−1.42 − 0.486i)10-s + 0.183i·11-s + 0.288i·13-s + (−0.545 + 1.59i)14-s + (0.253 + 0.967i)16-s + 0.743·17-s + 0.229i·19-s + (0.920 − 1.19i)20-s + (−0.173 − 0.0592i)22-s + 1.77·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.552 - 0.833i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -0.552 - 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.688435602\)
\(L(\frac12)\) \(\approx\) \(1.688435602\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.456 - 1.33i)T \)
3 \( 1 \)
19 \( 1 - iT \)
good5 \( 1 - 3.36iT - 5T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 - 0.608iT - 11T^{2} \)
13 \( 1 - 1.03iT - 13T^{2} \)
17 \( 1 - 3.06T + 17T^{2} \)
23 \( 1 - 8.50T + 23T^{2} \)
29 \( 1 + 7.27iT - 29T^{2} \)
31 \( 1 - 4.02T + 31T^{2} \)
37 \( 1 + 4.31iT - 37T^{2} \)
41 \( 1 - 4.15T + 41T^{2} \)
43 \( 1 - 6.27iT - 43T^{2} \)
47 \( 1 + 4.73T + 47T^{2} \)
53 \( 1 - 6.98iT - 53T^{2} \)
59 \( 1 + 2.64iT - 59T^{2} \)
61 \( 1 + 5.11iT - 61T^{2} \)
67 \( 1 - 2.62iT - 67T^{2} \)
71 \( 1 + 12.0T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + 0.913T + 79T^{2} \)
83 \( 1 - 0.887iT - 83T^{2} \)
89 \( 1 - 7.61T + 89T^{2} \)
97 \( 1 + 5.93T + 97T^{2} \)
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

−9.829952295198378976886896078729, −8.918427838840400683382638729631, −7.899062038576748159844693681775, −7.56915173335692635948729596981, −6.73350101908725261292425458004, −5.89109529474519085941415351941, −4.95931842998604120445446727255, −4.11715709202800775775940513808, −2.72711279409952339139388506392, −1.36976354048041330381686726233, 0.984067801459624489227430130069, 1.55321057336220789923495501003, 2.99526496935030018079538998125, 4.29806014936781506293359403689, 4.99530198697869347688022486177, 5.38501026197707412989570159822, 7.27438386119649877278135408478, 8.088711943890103187263968139991, 8.668653491550550256565433475769, 9.094549594116758007744364599872

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