↑

Properties

Label 2-28e2-16.5-c1-0-76
Degree $2$
Conductor $784$
Sign $-0.654 - 0.755i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
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  + (0.163 − 1.40i)2-s + (2.05 − 2.05i)3-s + (−1.94 − 0.459i)4-s + (−2.72 − 2.72i)5-s + (−2.55 − 3.22i)6-s + (−0.964 + 2.65i)8-s − 5.44i·9-s + (−4.27 + 3.38i)10-s + (0.919 + 0.919i)11-s + (−4.94 + 3.05i)12-s + (1.12 − 1.12i)13-s − 11.2·15-s + (3.57 + 1.78i)16-s + 1.50·17-s + (−7.65 − 0.891i)18-s + (−1.46 + 1.46i)19-s + ⋯
L(s)  = 1  + (0.115 − 0.993i)2-s + (1.18 − 1.18i)3-s + (−0.973 − 0.229i)4-s + (−1.21 − 1.21i)5-s + (−1.04 − 1.31i)6-s + (−0.340 + 0.940i)8-s − 1.81i·9-s + (−1.35 + 1.07i)10-s + (0.277 + 0.277i)11-s + (−1.42 + 0.881i)12-s + (0.312 − 0.312i)13-s − 2.89·15-s + (0.894 + 0.447i)16-s + 0.365·17-s + (−1.80 − 0.210i)18-s + (−0.335 + 0.335i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.654 - 0.755i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.638804 + 1.39904i\)
\(L(\frac12)\) \(\approx\) \(0.638804 + 1.39904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.163 + 1.40i)T \)
7 \( 1 \)
good3 \( 1 + (-2.05 + 2.05i)T - 3iT^{2} \)
5 \( 1 + (2.72 + 2.72i)T + 5iT^{2} \)
11 \( 1 + (-0.919 - 0.919i)T + 11iT^{2} \)
13 \( 1 + (-1.12 + 1.12i)T - 13iT^{2} \)
17 \( 1 - 1.50T + 17T^{2} \)
19 \( 1 + (1.46 - 1.46i)T - 19iT^{2} \)
23 \( 1 + 4.77iT - 23T^{2} \)
29 \( 1 + (-4.10 + 4.10i)T - 29iT^{2} \)
31 \( 1 + 4.10T + 31T^{2} \)
37 \( 1 + (1.65 + 1.65i)T + 37iT^{2} \)
41 \( 1 - 7.45iT - 41T^{2} \)
43 \( 1 + (-5.68 - 5.68i)T + 43iT^{2} \)
47 \( 1 + 3.59T + 47T^{2} \)
53 \( 1 + (0.675 + 0.675i)T + 53iT^{2} \)
59 \( 1 + (1.13 + 1.13i)T + 59iT^{2} \)
61 \( 1 + (-3.21 + 3.21i)T - 61iT^{2} \)
67 \( 1 + (1.52 - 1.52i)T - 67iT^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 + 1.77T + 79T^{2} \)
83 \( 1 + (-7.16 + 7.16i)T - 83iT^{2} \)
89 \( 1 + 8.45iT - 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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

−9.450610655793196924363484418570, −8.830617647948506065392446874727, −8.075901386059309800162239112159, −7.75569768918675115780729259523, −6.31052383389296835846557155833, −4.81623630718733842258378742713, −3.94688430544980908126740656769, −3.03092113934425044246645735378, −1.73445491856747751941634542474, −0.68484437299017415742448186954, 2.90980877520529859804689917116, 3.71429260275158351369784241553, 4.16457579841090324870239241343, 5.42610543134464683069614503386, 6.80938398585093317869690898639, 7.43600850768491443889281199566, 8.335992822682896295663327980448, 8.868010091025501783058942416753, 9.815815832707007038594087972930, 10.61693042248046239706175578475

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