↑

Properties

Label 2-896-16.13-c1-0-22
Degree $2$
Conductor $896$
Sign $-0.966 + 0.255i$
Analytic cond. $7.15459$
Root an. cond. $2.67480$
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  + (−0.416 − 0.416i)3-s + (1.13 − 1.13i)5-s − i·7-s − 2.65i·9-s + (−3.85 + 3.85i)11-s + (−4.66 − 4.66i)13-s − 0.943·15-s − 5.33·17-s + (2.55 + 2.55i)19-s + (−0.416 + 0.416i)21-s − 2.60i·23-s + 2.43i·25-s + (−2.35 + 2.35i)27-s + (1.22 + 1.22i)29-s − 0.833·31-s + ⋯
L(s)  = 1  + (−0.240 − 0.240i)3-s + (0.506 − 0.506i)5-s − 0.377i·7-s − 0.884i·9-s + (−1.16 + 1.16i)11-s + (−1.29 − 1.29i)13-s − 0.243·15-s − 1.29·17-s + (0.587 + 0.587i)19-s + (−0.0909 + 0.0909i)21-s − 0.543i·23-s + 0.487i·25-s + (−0.453 + 0.453i)27-s + (0.227 + 0.227i)29-s − 0.149·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.966 + 0.255i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.966 + 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0802964 - 0.617342i\)
\(L(\frac12)\) \(\approx\) \(0.0802964 - 0.617342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 + (0.416 + 0.416i)T + 3iT^{2} \)
5 \( 1 + (-1.13 + 1.13i)T - 5iT^{2} \)
11 \( 1 + (3.85 - 3.85i)T - 11iT^{2} \)
13 \( 1 + (4.66 + 4.66i)T + 13iT^{2} \)
17 \( 1 + 5.33T + 17T^{2} \)
19 \( 1 + (-2.55 - 2.55i)T + 19iT^{2} \)
23 \( 1 + 2.60iT - 23T^{2} \)
29 \( 1 + (-1.22 - 1.22i)T + 29iT^{2} \)
31 \( 1 + 0.833T + 31T^{2} \)
37 \( 1 + (-4.42 + 4.42i)T - 37iT^{2} \)
41 \( 1 + 0.263iT - 41T^{2} \)
43 \( 1 + (1.25 - 1.25i)T - 43iT^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + (0.0476 - 0.0476i)T - 53iT^{2} \)
59 \( 1 + (-3.60 + 3.60i)T - 59iT^{2} \)
61 \( 1 + (4.46 + 4.46i)T + 61iT^{2} \)
67 \( 1 + (9.50 + 9.50i)T + 67iT^{2} \)
71 \( 1 - 2.05iT - 71T^{2} \)
73 \( 1 + 5.48iT - 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 + (5.84 + 5.84i)T + 83iT^{2} \)
89 \( 1 + 6.32iT - 89T^{2} \)
97 \( 1 - 18.8T + 97T^{2} \)
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

−9.841797906007511766809063447047, −9.048527483714611079576382906718, −7.83580983777721958664617787170, −7.28460035950943057988443682991, −6.25175813763103573612439826798, −5.21978602817187976385959410626, −4.62596765885635843080040110481, −3.11370493978303163136696184335, −1.92781995635732902415428131231, −0.27782279428510401818457925851, 2.18776261667795780679215217808, 2.82970225131042238459282872330, 4.51515904037733806127833861041, 5.19138639208666956558766005775, 6.14447941906129963597227207133, 7.04411999281049180407063088906, 7.974594077117404365864674592001, 8.897734786550559185026836152692, 9.788192103029590388243958048738, 10.47253001231767589240737482524

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