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Properties

Label 2-1134-9.7-c1-0-9
Degree $2$
Conductor $1134$
Sign $0.173 - 0.984i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.5 + 2.59i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s − 3·10-s + (3 − 5.19i)11-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 3·17-s + 2·19-s + (1.50 − 2.59i)20-s + (3 + 5.19i)22-s + (3 + 5.19i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.188 + 0.327i)7-s + 0.353·8-s − 0.948·10-s + (0.904 − 1.56i)11-s + (0.138 + 0.240i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s + 0.727·17-s + 0.458·19-s + (0.335 − 0.580i)20-s + (0.639 + 1.10i)22-s + (0.625 + 1.08i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.607722610\)
\(L(\frac12)\) \(\approx\) \(1.607722610\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.959412636747884270068846904767, −9.102048515020655962584395839390, −8.456095146412158674561356909972, −7.38635002088883832413786120331, −6.52971027584129180400570239012, −6.06612286073431944210998771369, −5.23040478913913337874608773361, −3.62751817610428634708668978409, −2.85175169230445470170579818927, −1.25317574174354436405884938233, 0.997023425202170479832554010303, 1.87430426293817310268732094758, 3.27536199925732988859333826106, 4.53479084820914783957455729668, 5.00854605935149852447247633123, 6.34165974762890287419742700325, 7.21732197553549228913214154370, 8.259221759748213239029838352112, 8.996558239787216145534504043210, 9.793009425611540517045828265720

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