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Properties

Label 2-1134-9.4-c1-0-4
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 + (1.5 + 2.59i)11-s + (2 − 3.46i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 6·17-s − 7·19-s + (−1.50 − 2.59i)20-s + (1.5 − 2.59i)22-s + (−1.5 + 2.59i)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 + (0.452 + 0.783i)11-s + (0.554 − 0.960i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + 1.45·17-s − 1.60·19-s + (−0.335 − 0.580i)20-s + (0.319 − 0.553i)22-s + (−0.312 + 0.541i)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: $\chi_{1134} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6925010458\)
\(L(\frac12)\) \(\approx\) \(0.6925010458\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.21307271205712844094756177272, −9.468783435582933890938923726914, −8.277300945373504316623385076704, −7.68730105763419994096189796142, −6.89048029226982556013977877695, −5.99419603552723828146712530956, −4.57150825542166129561227477087, −3.55058116213267723387501508738, −3.00902470852328674728299248161, −1.50210857448012576323703618716, 0.35845510065607159860200186987, 1.75087567053039235246962591562, 3.65195907699735958839635850664, 4.37128042795252183440702999653, 5.47568081231923783502518735379, 6.18746122893249894922842204710, 7.15597434769857276055366323903, 8.220710847916428281024307274536, 8.682221572666884403664101095894, 9.180688730585638236743667316358

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