↑

Properties

Label 2-1134-63.25-c1-0-3
Degree $2$
Conductor $1134$
Sign $0.585 - 0.810i$
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  − 2-s + 4-s + (−2.62 + 0.358i)7-s − 8-s + (−2.12 − 3.67i)11-s + (1.12 + 1.94i)13-s + (2.62 − 0.358i)14-s + 16-s + (1.12 + 1.94i)19-s + (2.12 + 3.67i)22-s + (−0.621 + 1.07i)23-s + (2.5 + 4.33i)25-s + (−1.12 − 1.94i)26-s + (−2.62 + 0.358i)28-s + (2.12 − 3.67i)29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.990 + 0.135i)7-s − 0.353·8-s + (−0.639 − 1.10i)11-s + (0.310 + 0.538i)13-s + (0.700 − 0.0958i)14-s + 0.250·16-s + (0.257 + 0.445i)19-s + (0.452 + 0.783i)22-s + (−0.129 + 0.224i)23-s + (0.5 + 0.866i)25-s + (−0.219 − 0.380i)26-s + (−0.495 + 0.0677i)28-s + (0.393 − 0.682i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.585 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8424291597\)
\(L(\frac12)\) \(\approx\) \(0.8424291597\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 + (2.62 - 0.358i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.12 + 3.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.12 - 1.94i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.12 - 1.94i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.621 - 1.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.12 + 3.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.24T + 31T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.74 - 9.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.24 - 9.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.75T + 47T^{2} \)
53 \( 1 + (2.12 - 3.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2.24T + 61T^{2} \)
67 \( 1 - 0.242T + 67T^{2} \)
71 \( 1 + 1.24T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.757T + 79T^{2} \)
83 \( 1 + (-8.12 + 14.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.74 - 9.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.24 - 3.88i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.810972429363899442158760222063, −9.230488728019413895210086117473, −8.283079058547677689581959094703, −7.71141285746211010885598933677, −6.36585166992815776476110421962, −6.18742400007649537410871618497, −4.83470861529287452892331775937, −3.42096954632153069693973601571, −2.71444585009447647173835229913, −1.05829319207624412608838134597, 0.56026880286845645292648662285, 2.27184772789444736902443874373, 3.18599299319625536952875040931, 4.46787822466665036350590052947, 5.56980416333976354319365072486, 6.61812369212987268651370680819, 7.17196414325511648224557787967, 8.134198175471581296168146362528, 8.898105893728767224521319820606, 9.860467720890095960511629410799

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