↑

Properties

Label 2-946-473.472-c1-0-3
Degree $2$
Conductor $946$
Sign $-0.925 + 0.379i$
Analytic cond. $7.55384$
Root an. cond. $2.74842$
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  + 2-s + 2.75i·3-s + 4-s − 1.19i·5-s + 2.75i·6-s − 5.12·7-s + 8-s − 4.60·9-s − 1.19i·10-s + (−1.29 + 3.05i)11-s + 2.75i·12-s − 4.04i·13-s − 5.12·14-s + 3.28·15-s + 16-s + 3.50i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.59i·3-s + 0.5·4-s − 0.533i·5-s + 1.12i·6-s − 1.93·7-s + 0.353·8-s − 1.53·9-s − 0.377i·10-s + (−0.389 + 0.921i)11-s + 0.795i·12-s − 1.12i·13-s − 1.36·14-s + 0.849·15-s + 0.250·16-s + 0.850i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(946\)    =    \(2 \cdot 11 \cdot 43\)
Sign: $-0.925 + 0.379i$
Analytic conductor: \(7.55384\)
Root analytic conductor: \(2.74842\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{946} (945, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 946,\ (\ :1/2),\ -0.925 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.136995 - 0.694399i\)
\(L(\frac12)\) \(\approx\) \(0.136995 - 0.694399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
11 \( 1 + (1.29 - 3.05i)T \)
43 \( 1 + (-4.65 - 4.61i)T \)
good3 \( 1 - 2.75iT - 3T^{2} \)
5 \( 1 + 1.19iT - 5T^{2} \)
7 \( 1 + 5.12T + 7T^{2} \)
13 \( 1 + 4.04iT - 13T^{2} \)
17 \( 1 - 3.50iT - 17T^{2} \)
19 \( 1 + 6.35T + 19T^{2} \)
23 \( 1 + 1.53T + 23T^{2} \)
29 \( 1 + 8.18T + 29T^{2} \)
31 \( 1 + 1.26T + 31T^{2} \)
37 \( 1 - 2.43iT - 37T^{2} \)
41 \( 1 + 0.736iT - 41T^{2} \)
47 \( 1 + 7.13T + 47T^{2} \)
53 \( 1 - 1.01T + 53T^{2} \)
59 \( 1 - 8.91T + 59T^{2} \)
61 \( 1 + 1.09T + 61T^{2} \)
67 \( 1 + 1.54T + 67T^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 + 10.0iT - 79T^{2} \)
83 \( 1 - 5.77iT - 83T^{2} \)
89 \( 1 - 16.7iT - 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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

−10.38208377643787592073082241027, −9.846612354380784815331626056640, −9.163104630513661716141163302332, −8.164780942275611634501766871848, −6.83426920829290771760608320079, −5.92126560033947186863230141580, −5.19210910517198911132560523310, −4.15923821459955844083844554347, −3.59473049874760389943731107891, −2.57923994241790657617448087732, 0.23434107835801840804662356378, 2.13791311173480933149086282701, 2.93740994165651064845622617454, 3.89035320144487255399061372975, 5.62406972907062096299333242090, 6.39495579135721262787394556634, 6.77408578593810677031318655608, 7.43712692698257367211335307435, 8.645744474524748375968593771378, 9.510767194586234575708086150732

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