↑

Properties

Label 2-28e2-112.109-c1-0-28
Degree $2$
Conductor $784$
Sign $0.300 - 0.953i$
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  + (0.262 + 1.38i)2-s + (−2.91 + 0.781i)3-s + (−1.86 + 0.729i)4-s + (0.745 + 0.199i)5-s + (−1.85 − 3.84i)6-s + (−1.50 − 2.39i)8-s + (5.28 − 3.05i)9-s + (−0.0820 + 1.08i)10-s + (−0.333 − 1.24i)11-s + (4.85 − 3.57i)12-s + (0.919 + 0.919i)13-s − 2.32·15-s + (2.93 − 2.71i)16-s + (3.95 − 6.85i)17-s + (5.63 + 6.54i)18-s + (−0.478 + 1.78i)19-s + ⋯
L(s)  = 1  + (0.185 + 0.982i)2-s + (−1.68 + 0.450i)3-s + (−0.931 + 0.364i)4-s + (0.333 + 0.0893i)5-s + (−0.755 − 1.57i)6-s + (−0.530 − 0.847i)8-s + (1.76 − 1.01i)9-s + (−0.0259 + 0.344i)10-s + (−0.100 − 0.374i)11-s + (1.40 − 1.03i)12-s + (0.254 + 0.254i)13-s − 0.601·15-s + (0.734 − 0.678i)16-s + (0.959 − 1.66i)17-s + (1.32 + 1.54i)18-s + (−0.109 + 0.409i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692115 + 0.507658i\)
\(L(\frac12)\) \(\approx\) \(0.692115 + 0.507658i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.262 - 1.38i)T \)
7 \( 1 \)
good3 \( 1 + (2.91 - 0.781i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.745 - 0.199i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.333 + 1.24i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.919 - 0.919i)T + 13iT^{2} \)
17 \( 1 + (-3.95 + 6.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.478 - 1.78i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.33 + 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.25 - 5.25i)T + 29iT^{2} \)
31 \( 1 + (2.44 - 4.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.28 + 0.343i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.84iT - 41T^{2} \)
43 \( 1 + (-0.585 + 0.585i)T - 43iT^{2} \)
47 \( 1 + (-2.86 - 4.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.54 - 9.51i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.39 + 8.93i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.71 + 6.38i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-5.94 + 1.59i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.99iT - 71T^{2} \)
73 \( 1 + (-6.69 - 3.86i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.63 + 8.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.78 - 4.78i)T + 83iT^{2} \)
89 \( 1 + (-1.84 + 1.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.01T + 97T^{2} \)
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

−10.41178627630864870969509687981, −9.692380942669384479334584110704, −8.828073174162162860800796574553, −7.53206816489312844626990973878, −6.72963368347010615558993312048, −5.99744110727210463394661929125, −5.23937833309784082504392947448, −4.65634469173750829201774894163, −3.37029385933612182749870474899, −0.75375365525738410209878802268, 0.913371554138145586908841095614, 2.00630070460261099463326454831, 3.73978801076530126299655348462, 4.81246973019107103321816139729, 5.68278771351034273963874730811, 6.16949439288336510248810865950, 7.44192749180644954973961948099, 8.509671065222664807476149423291, 9.845881303951246518186407439207, 10.27223469428863710884378211740

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