↑

Properties

Label 2-840-280.139-c1-0-85
Degree $2$
Conductor $840$
Sign $0.473 + 0.880i$
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  + (1.31 − 0.512i)2-s + 3-s + (1.47 − 1.35i)4-s + (1.36 − 1.77i)5-s + (1.31 − 0.512i)6-s + (−2.64 + 0.0973i)7-s + (1.24 − 2.53i)8-s + 9-s + (0.891 − 3.03i)10-s + 5.35·11-s + (1.47 − 1.35i)12-s + 6.03i·13-s + (−3.43 + 1.48i)14-s + (1.36 − 1.77i)15-s + (0.343 − 3.98i)16-s − 3.08·17-s + ⋯
L(s)  = 1  + (0.931 − 0.362i)2-s + 0.577·3-s + (0.736 − 0.676i)4-s + (0.610 − 0.791i)5-s + (0.538 − 0.209i)6-s + (−0.999 + 0.0367i)7-s + (0.441 − 0.897i)8-s + 0.333·9-s + (0.281 − 0.959i)10-s + 1.61·11-s + (0.425 − 0.390i)12-s + 1.67i·13-s + (−0.917 + 0.396i)14-s + (0.352 − 0.457i)15-s + (0.0858 − 0.996i)16-s − 0.747·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.473 + 0.880i$
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: \((2,\ 840,\ (\ :1/2),\ 0.473 + 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.01633 - 1.80287i\)
\(L(\frac12)\) \(\approx\) \(3.01633 - 1.80287i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 + 0.512i)T \)
3 \( 1 - T \)
5 \( 1 + (-1.36 + 1.77i)T \)
7 \( 1 + (2.64 - 0.0973i)T \)
good11 \( 1 - 5.35T + 11T^{2} \)
13 \( 1 - 6.03iT - 13T^{2} \)
17 \( 1 + 3.08T + 17T^{2} \)
19 \( 1 + 0.109iT - 19T^{2} \)
23 \( 1 - 2.14T + 23T^{2} \)
29 \( 1 + 7.09iT - 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 4.52T + 37T^{2} \)
41 \( 1 + 4.77iT - 41T^{2} \)
43 \( 1 - 6.25iT - 43T^{2} \)
47 \( 1 - 4.11iT - 47T^{2} \)
53 \( 1 + 3.67T + 53T^{2} \)
59 \( 1 - 12.0iT - 59T^{2} \)
61 \( 1 - 8.16T + 61T^{2} \)
67 \( 1 - 7.52iT - 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 3.38T + 73T^{2} \)
79 \( 1 - 13.9iT - 79T^{2} \)
83 \( 1 - 3.39T + 83T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 + 6.53T + 97T^{2} \)
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

−9.776772758695135111185439191598, −9.352826887041471475845632682859, −8.780966909666030833065281894558, −7.01348058953145200167829101549, −6.57178293023714481732166449112, −5.63650996797990192888164277588, −4.26994497808735394606381319856, −3.89955776002573291262437459455, −2.41467833993169844564688518751, −1.44642266728296568682168101152, 1.97590624576057310857044898179, 3.37552162962545268237528797341, 3.47847538495441960746195692114, 5.12183888150493958455772136346, 6.11624000236184279843753182877, 6.79072393941312608334749876830, 7.40032920689508941567835208922, 8.699402614205786302092904809119, 9.419637975880277954921891386850, 10.47100740520948188878154451129

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