↑

Properties

Label 2-840-105.89-c1-0-34
Degree $2$
Conductor $840$
Sign $0.749 + 0.662i$
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  + (1.58 − 0.703i)3-s + (−0.851 + 2.06i)5-s + (−1.08 − 2.41i)7-s + (2.01 − 2.22i)9-s + (3.40 − 1.96i)11-s − 2.95·13-s + (0.106 + 3.87i)15-s + (2.29 − 1.32i)17-s + (6.11 + 3.53i)19-s + (−3.41 − 3.05i)21-s + (4.18 − 7.25i)23-s + (−3.54 − 3.52i)25-s + (1.61 − 4.93i)27-s − 1.79i·29-s + (−5.96 + 3.44i)31-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)3-s + (−0.380 + 0.924i)5-s + (−0.409 − 0.912i)7-s + (0.670 − 0.742i)9-s + (1.02 − 0.592i)11-s − 0.818·13-s + (0.0274 + 0.999i)15-s + (0.556 − 0.321i)17-s + (1.40 + 0.810i)19-s + (−0.745 − 0.666i)21-s + (0.873 − 1.51i)23-s + (−0.709 − 0.704i)25-s + (0.310 − 0.950i)27-s − 0.334i·29-s + (−1.07 + 0.618i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89383 - 0.717208i\)
\(L(\frac12)\) \(\approx\) \(1.89383 - 0.717208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.58 + 0.703i)T \)
5 \( 1 + (0.851 - 2.06i)T \)
7 \( 1 + (1.08 + 2.41i)T \)
good11 \( 1 + (-3.40 + 1.96i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.95T + 13T^{2} \)
17 \( 1 + (-2.29 + 1.32i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.11 - 3.53i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.18 + 7.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.79iT - 29T^{2} \)
31 \( 1 + (5.96 - 3.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.34 - 4.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.41T + 41T^{2} \)
43 \( 1 + 4.62iT - 43T^{2} \)
47 \( 1 + (-1.77 - 1.02i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.01 + 1.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.29 - 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.88 + 2.81i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.62 - 4.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.62iT - 71T^{2} \)
73 \( 1 + (4.14 + 7.17i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.84 - 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.53iT - 83T^{2} \)
89 \( 1 + (4.32 - 7.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.65T + 97T^{2} \)
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

−9.969666262689850189799754531270, −9.366953746744278075907242173005, −8.274473026489186275278159971596, −7.34931967140453452063048738966, −7.01817327665786246971841737446, −6.02182594724767962595550088573, −4.34263181805435769568132146879, −3.45297429420785651793449356238, −2.77059407083115420529222652784, −1.03625702854863065904487916537, 1.51837562991743584556234750511, 2.88770327089886147272191108010, 3.86619894065121789177386499056, 4.88770707023359541565779579504, 5.65921198887050911212457560277, 7.24091191608784908911408430581, 7.72268310058933386691701285797, 8.954685198380249158542611287597, 9.370849602202689593227263747250, 9.737636140856474759828428234865

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