Properties

Label 2-1134-63.4-c1-0-28
Degree $2$
Conductor $1134$
Sign $-0.0788 + 0.996i$
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 + 3·5-s + (−2 − 1.73i)7-s − 0.999·8-s + (1.5 − 2.59i)10-s + 3·11-s + (2 − 3.46i)13-s + (−2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)19-s + (−1.49 − 2.59i)20-s + (1.5 − 2.59i)22-s + 4·25-s + (−1.99 − 3.46i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.34·5-s + (−0.755 − 0.654i)7-s − 0.353·8-s + (0.474 − 0.821i)10-s + 0.904·11-s + (0.554 − 0.960i)13-s + (−0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.458 + 0.794i)19-s + (−0.335 − 0.580i)20-s + (0.319 − 0.553i)22-s + 0.800·25-s + (−0.392 − 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.0788 + 0.996i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.0788 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.310142906\)
\(L(\frac12)\) \(\approx\) \(2.310142906\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 1.73i)T \)
good5 \( 1 - 3T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
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 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.708486461572133428219338457256, −9.226637517113060471890790531188, −8.024947940780678758494062239078, −6.85418599755038843685713188580, −5.98305633488629024911144092968, −5.54156348627222818506458121028, −4.10870492650767046906267084903, −3.35661054123594239836902458680, −2.14775661198896399236058700745, −0.977203350482848606751121787607, 1.62767528143009047053710113133, 2.85087939988167470254451959724, 3.95234891995927076790138491199, 5.17560042390610385257426363034, 5.89224375491954948816210244283, 6.57932104836268738520748312337, 7.15498269762732117768440445654, 8.739549406789580451893183727713, 9.114712235258629449525666312509, 9.691122025135894266960511988428

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