↑

Properties

Label 2-840-280.139-c1-0-14
Degree $2$
Conductor $840$
Sign $0.480 - 0.876i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
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  + (0.355 − 1.36i)2-s + 3-s + (−1.74 − 0.974i)4-s + (−1.14 + 1.92i)5-s + (0.355 − 1.36i)6-s + (1.89 + 1.84i)7-s + (−1.95 + 2.04i)8-s + 9-s + (2.22 + 2.24i)10-s − 6.38·11-s + (−1.74 − 0.974i)12-s + 1.62i·13-s + (3.20 − 1.93i)14-s + (−1.14 + 1.92i)15-s + (2.10 + 3.40i)16-s − 3.66·17-s + ⋯
L(s)  = 1  + (0.251 − 0.967i)2-s + 0.577·3-s + (−0.873 − 0.487i)4-s + (−0.511 + 0.859i)5-s + (0.145 − 0.558i)6-s + (0.716 + 0.697i)7-s + (−0.691 + 0.722i)8-s + 0.333·9-s + (0.702 + 0.711i)10-s − 1.92·11-s + (−0.504 − 0.281i)12-s + 0.449i·13-s + (0.855 − 0.517i)14-s + (−0.295 + 0.496i)15-s + (0.525 + 0.850i)16-s − 0.889·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.480 - 0.876i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.480 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966248 + 0.572125i\)
\(L(\frac12)\) \(\approx\) \(0.966248 + 0.572125i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.355 + 1.36i)T \)
3 \( 1 - T \)
5 \( 1 + (1.14 - 1.92i)T \)
7 \( 1 + (-1.89 - 1.84i)T \)
good11 \( 1 + 6.38T + 11T^{2} \)
13 \( 1 - 1.62iT - 13T^{2} \)
17 \( 1 + 3.66T + 17T^{2} \)
19 \( 1 - 5.64iT - 19T^{2} \)
23 \( 1 + 0.543T + 23T^{2} \)
29 \( 1 - 6.72iT - 29T^{2} \)
31 \( 1 - 4.92T + 31T^{2} \)
37 \( 1 + 8.14T + 37T^{2} \)
41 \( 1 + 2.75iT - 41T^{2} \)
43 \( 1 + 0.448iT - 43T^{2} \)
47 \( 1 - 0.873iT - 47T^{2} \)
53 \( 1 - 9.55T + 53T^{2} \)
59 \( 1 + 2.84iT - 59T^{2} \)
61 \( 1 - 7.86T + 61T^{2} \)
67 \( 1 + 3.43iT - 67T^{2} \)
71 \( 1 - 5.58iT - 71T^{2} \)
73 \( 1 - 7.80T + 73T^{2} \)
79 \( 1 - 1.92iT - 79T^{2} \)
83 \( 1 - 7.30T + 83T^{2} \)
89 \( 1 + 3.48iT - 89T^{2} \)
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

−10.58027050640711178506467165180, −9.738534019535998231902574714269, −8.486415965395889195044269681336, −8.196625254663454701819270253244, −7.04649657085324126412286144888, −5.67336534853008994413832424535, −4.81958796137333874567639451625, −3.72093507601378753112487038722, −2.70472285279820254220146498358, −2.00283324422655736149279744161, 0.45263467222329508628555399569, 2.60629605088839914764782304988, 3.99144493791082803735192957419, 4.80431468115769805071554693765, 5.35800735902905403772948057202, 6.86060459241063420844990829805, 7.69686713585880114701305553693, 8.179724440613199297117007688574, 8.783433684821288975272048419651, 9.883487313991184508042552209683

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