↑

Properties

Label 2-1134-63.38-c1-0-10
Degree $2$
Conductor $1134$
Sign $0.818 - 0.574i$
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 + (−1.22 + 2.12i)5-s + (−1.62 − 2.09i)7-s + i·8-s + (2.12 + 1.22i)10-s + (3.67 − 2.12i)11-s + (−0.621 + 0.358i)13-s + (−2.09 + 1.62i)14-s + 16-s + (−1.22 + 2.12i)17-s + (−4.24 + 2.44i)19-s + (1.22 − 2.12i)20-s + (−2.12 − 3.67i)22-s + (5.19 + 3i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.547 + 0.948i)5-s + (−0.612 − 0.790i)7-s + 0.353i·8-s + (0.670 + 0.387i)10-s + (1.10 − 0.639i)11-s + (−0.172 + 0.0994i)13-s + (−0.558 + 0.433i)14-s + 0.250·16-s + (−0.297 + 0.514i)17-s + (−0.973 + 0.561i)19-s + (0.273 − 0.474i)20-s + (−0.452 − 0.783i)22-s + (1.08 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.818 - 0.574i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.818 - 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.031160838\)
\(L(\frac12)\) \(\approx\) \(1.031160838\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.621 - 0.358i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.22 - 2.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.24 - 2.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.52 + 0.878i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.08iT - 31T^{2} \)
37 \( 1 + (2.62 + 4.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.22 - 2.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + (-12.5 - 7.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.44T + 59T^{2} \)
61 \( 1 - 4.18iT - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (4.75 + 2.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.757T + 79T^{2} \)
83 \( 1 + (7.64 - 13.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.52 + 2.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.74 + 1.58i)T + (48.5 + 84.0i)T^{2} \)
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

−10.11825261947084913784028391982, −9.155222651944567145822580628701, −8.449355886271992172813827854965, −7.21138688199879366576397763161, −6.76433265730334975818035080600, −5.71331026461551730525907935975, −4.18340892436498822396267130306, −3.69104136712967956785736267832, −2.80912391257787855643239534409, −1.22544189118304985333708937457, 0.51718394439874704471367590391, 2.33451403757104796606720910326, 3.85089260871354659326795005598, 4.62006759821355296728877910850, 5.45016219190110289359762969243, 6.53804135304181830241361335707, 7.08337740637392311071200272657, 8.229679727668566399662068884692, 9.026664487168148581325507213066, 9.214157703296154639252955285803

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