↑

Properties

Label 2-1134-9.4-c1-0-9
Degree $2$
Conductor $1134$
Sign $0.984 - 0.173i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.133 − 0.232i)5-s + (0.5 + 0.866i)7-s + 0.999·8-s − 0.267·10-s + (3.09 + 5.36i)11-s + (3.23 − 5.59i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 7·17-s + 0.732·19-s + (0.133 + 0.232i)20-s + (3.09 − 5.36i)22-s + (−2.09 + 3.63i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0599 − 0.103i)5-s + (0.188 + 0.327i)7-s + 0.353·8-s − 0.0847·10-s + (0.934 + 1.61i)11-s + (0.896 − 1.55i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s − 1.69·17-s + 0.167·19-s + (0.0299 + 0.0518i)20-s + (0.660 − 1.14i)22-s + (−0.437 + 0.757i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.357477196\)
\(L(\frac12)\) \(\approx\) \(1.357477196\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.133 + 0.232i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.09 - 5.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.23 + 5.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 - 0.732T + 19T^{2} \)
23 \( 1 + (2.09 - 3.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.767 - 1.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.09 - 7.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + (-1.26 + 2.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.732 - 1.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.36 - 4.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.46T + 53T^{2} \)
59 \( 1 + (-2.09 + 3.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.96 + 3.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.36 + 5.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.53T + 71T^{2} \)
73 \( 1 - 8.26T + 73T^{2} \)
79 \( 1 + (-4.56 - 7.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.29 - 14.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 + (5.46 + 9.46i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.717353534381979546323565137891, −9.173939464050889866861669219350, −8.393170944517625606803267129167, −7.46002344391935888848358835982, −6.62858979995626503448401987891, −5.46795525338221720890159210045, −4.50254309061703550009047879704, −3.56303409935884354290170176006, −2.33026107097674820849612917685, −1.26416446183049235854952924082, 0.78077031696565700672146600386, 2.24851710181509969617105862254, 3.93823311093402333688561153616, 4.42175339431111310602425777761, 6.05988477447521291914017864234, 6.31089670180172799703222396171, 7.18576866068511917622227116717, 8.425232491173876032374011154384, 8.787885257302960281269941369911, 9.497267558844951810179299782618

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