↑

Properties

Label 2-896-112.75-c1-0-14
Degree $2$
Conductor $896$
Sign $0.474 - 0.880i$
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  + (2.29 + 0.615i)3-s + (0.223 + 0.835i)5-s + (1.25 + 2.32i)7-s + (2.29 + 1.32i)9-s + (−2.03 − 0.544i)11-s + (0.336 + 0.336i)13-s + 2.05i·15-s + (−0.0488 + 0.0282i)17-s + (2.11 + 7.89i)19-s + (1.45 + 6.11i)21-s + (3.59 − 6.23i)23-s + (3.68 − 2.12i)25-s + (−0.584 − 0.584i)27-s + (−4.81 + 4.81i)29-s + (1.84 + 3.20i)31-s + ⋯
L(s)  = 1  + (1.32 + 0.355i)3-s + (0.100 + 0.373i)5-s + (0.475 + 0.879i)7-s + (0.765 + 0.442i)9-s + (−0.613 − 0.164i)11-s + (0.0933 + 0.0933i)13-s + 0.531i·15-s + (−0.0118 + 0.00684i)17-s + (0.485 + 1.81i)19-s + (0.317 + 1.33i)21-s + (0.750 − 1.29i)23-s + (0.736 − 0.425i)25-s + (−0.112 − 0.112i)27-s + (−0.894 + 0.894i)29-s + (0.332 + 0.575i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.474 - 0.880i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.474 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16433 + 1.29210i\)
\(L(\frac12)\) \(\approx\) \(2.16433 + 1.29210i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 2.32i)T \)
good3 \( 1 + (-2.29 - 0.615i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.223 - 0.835i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.03 + 0.544i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.336 - 0.336i)T + 13iT^{2} \)
17 \( 1 + (0.0488 - 0.0282i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.11 - 7.89i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.59 + 6.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.81 - 4.81i)T - 29iT^{2} \)
31 \( 1 + (-1.84 - 3.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.27 - 0.876i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.95T + 41T^{2} \)
43 \( 1 + (-4.99 + 4.99i)T - 43iT^{2} \)
47 \( 1 + (-5.50 + 9.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.971 + 3.62i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.27 - 8.49i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.21 - 1.66i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.80 + 10.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 + (2.69 + 4.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.99 + 2.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.52 + 4.52i)T - 83iT^{2} \)
89 \( 1 + (-1.28 + 2.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.06iT - 97T^{2} \)
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

−10.27105669158036909349450851844, −9.192062115903167311150430800180, −8.594761774655174358625369818729, −8.044766163621919047273560767185, −7.03452283304873604324405078355, −5.83050419603318330881308979515, −4.91356439852601731627342378792, −3.63497410339652053860185035654, −2.84564004041085698320001067069, −1.89802201840283376264085324971, 1.13406337832797585552241220543, 2.43977143863737962292293279868, 3.38234261162738593958209409772, 4.52910502808123777092662401423, 5.42769382251775142538668370624, 6.99267427555479258186154142205, 7.53417036361114391354457889190, 8.199112104423933185929582273061, 9.184934652480611181783997414098, 9.594592557028375306323525812320

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