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Properties

Label 2-1134-21.17-c1-0-10
Degree $2$
Conductor $1134$
Sign $0.996 - 0.0808i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.330 + 0.571i)5-s + (−2.34 + 1.21i)7-s + 0.999i·8-s + (−0.571 − 0.330i)10-s + (2.25 + 1.30i)11-s − 5.87i·13-s + (1.42 − 2.22i)14-s + (−0.5 − 0.866i)16-s + (2.35 − 4.08i)17-s + (−3.59 + 2.07i)19-s + 0.660·20-s − 2.60·22-s + (4.56 − 2.63i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.147 + 0.255i)5-s + (−0.888 + 0.459i)7-s + 0.353i·8-s + (−0.180 − 0.104i)10-s + (0.678 + 0.392i)11-s − 1.62i·13-s + (0.381 − 0.595i)14-s + (−0.125 − 0.216i)16-s + (0.571 − 0.989i)17-s + (−0.824 + 0.475i)19-s + 0.147·20-s − 0.554·22-s + (0.951 − 0.549i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.996 - 0.0808i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.996 - 0.0808i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.116272309\)
\(L(\frac12)\) \(\approx\) \(1.116272309\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (2.34 - 1.21i)T \)
good5 \( 1 + (-0.330 - 0.571i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.25 - 1.30i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.87iT - 13T^{2} \)
17 \( 1 + (-2.35 + 4.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.59 - 2.07i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.56 + 2.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.43iT - 29T^{2} \)
31 \( 1 + (1.73 + 1.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.17 - 5.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.90T + 41T^{2} \)
43 \( 1 + 2.66T + 43T^{2} \)
47 \( 1 + (-0.874 - 1.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.83 - 4.52i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.111 - 0.193i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.78 + 4.49i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.67 + 9.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.89iT - 71T^{2} \)
73 \( 1 + (3.46 + 2.00i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.08 - 8.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + (7.52 + 13.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.33iT - 97T^{2} \)
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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

−9.791776646537198434447567296329, −9.027508716967403424738195927817, −8.272516675704569624583867273815, −7.29026604548689179616978044995, −6.57001395742021816703561006594, −5.80537921138023231813012759520, −4.87127136667372294822454590901, −3.35531833977992055940259775212, −2.51973931061024706715730234955, −0.790443847870108281787902039344, 1.00601636776239015735004265159, 2.26684409219741247989526825242, 3.64270221577690257635314592658, 4.24247368923131037904995220698, 5.76922418414892560854985157039, 6.65680052749045254675318708001, 7.23356515840505998496970990881, 8.427191875300766837292298978383, 9.203676264207622465676151956320, 9.566545718021393099290677548602

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