↑

Properties

Label 2-1134-63.38-c1-0-3
Degree $2$
Conductor $1134$
Sign $-0.942 + 0.333i$
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  + i·2-s − 4-s + (−0.358 + 0.621i)5-s + (−1.62 + 2.09i)7-s − i·8-s + (−0.621 − 0.358i)10-s + (2.59 − 1.5i)11-s + (2.12 − 1.22i)13-s + (−2.09 − 1.62i)14-s + 16-s + (−2.95 + 5.12i)17-s + (−5.12 + 2.95i)19-s + (0.358 − 0.621i)20-s + (1.5 + 2.59i)22-s + (−3.67 − 2.12i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.160 + 0.277i)5-s + (−0.612 + 0.790i)7-s − 0.353i·8-s + (−0.196 − 0.113i)10-s + (0.783 − 0.452i)11-s + (0.588 − 0.339i)13-s + (−0.558 − 0.433i)14-s + 0.250·16-s + (−0.717 + 1.24i)17-s + (−1.17 + 0.678i)19-s + (0.0802 − 0.138i)20-s + (0.319 + 0.553i)22-s + (−0.766 − 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.942 + 0.333i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6236062152\)
\(L(\frac12)\) \(\approx\) \(0.6236062152\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (1.62 - 2.09i)T \)
good5 \( 1 + (0.358 - 0.621i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.95 - 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.12 - 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.67 + 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.27 + 3.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.08iT - 31T^{2} \)
37 \( 1 + (-0.121 - 0.210i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.91 + 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.121 + 0.210i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.91T + 47T^{2} \)
53 \( 1 + (6.27 + 3.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.06T + 59T^{2} \)
61 \( 1 - 1.01iT - 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + (1.24 + 0.717i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 2.75T + 79T^{2} \)
83 \( 1 + (3.31 - 5.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.7 - 6.77i)T + (48.5 + 84.0i)T^{2} \)
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

−10.31769347132207543881006587764, −9.017486093940681126279305136471, −8.759751641310679243394893590498, −7.87680120301141299127158645512, −6.65153769090772866975310938585, −6.24096664530011980507944962184, −5.47031045336014324791961332820, −4.07324630756046056310836385451, −3.41739331920699072155651234410, −1.86006726308929513699459244382, 0.26691717899503167749137594922, 1.72719958176330372359363085636, 3.01185634507814845433919313960, 4.19313349780259626265242378850, 4.52979142696598385393698122884, 6.06972096908060777739871437071, 6.83705851051738531965108076027, 7.73687875604383313612902393016, 8.895956230998364219930113928632, 9.364091152904634812098539877889

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