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Properties

Label 2-1134-7.4-c1-0-20
Degree $2$
Conductor $1134$
Sign $0.996 + 0.0884i$
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 + (−0.296 − 0.514i)5-s + (−2.25 − 1.38i)7-s − 0.999·8-s + (0.296 − 0.514i)10-s + (0.296 − 0.514i)11-s + 2.51·13-s + (0.0665 − 2.64i)14-s + (−0.5 − 0.866i)16-s + (1.46 − 2.52i)17-s + (2.69 + 4.66i)19-s + 0.593·20-s + 0.593·22-s + (−2.23 − 3.86i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.132 − 0.229i)5-s + (−0.853 − 0.521i)7-s − 0.353·8-s + (0.0938 − 0.162i)10-s + (0.0894 − 0.154i)11-s + 0.697·13-s + (0.0177 − 0.706i)14-s + (−0.125 − 0.216i)16-s + (0.354 − 0.613i)17-s + (0.617 + 1.06i)19-s + 0.132·20-s + 0.126·22-s + (−0.465 − 0.805i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.996 + 0.0884i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.996 + 0.0884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.651393389\)
\(L(\frac12)\) \(\approx\) \(1.651393389\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 + 1.38i)T \)
good5 \( 1 + (0.296 + 0.514i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.296 + 0.514i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.51T + 13T^{2} \)
17 \( 1 + (-1.46 + 2.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 + 3.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 + (-3.93 + 6.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.273T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.02 + 6.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.32 - 7.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.32 - 5.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.956 + 1.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (-3.95 + 6.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.62 - 8.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.70T + 83T^{2} \)
89 \( 1 + (6.21 + 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−9.874000343970121818066569668064, −8.796848245954529336974735219293, −8.108656681859251025790753895704, −7.24168805466470909758783227495, −6.36000368141581235841522009553, −5.77168941788325730580137908450, −4.53648705930574021266879316607, −3.78754294841120163808914359849, −2.76255528833820574952490688770, −0.75969429186910760292145744173, 1.23765875919223297204159804616, 2.76960429527432855934885450836, 3.38746376391535684542032113688, 4.50663271225465705025083967029, 5.57715393409149301612759922136, 6.33290078135818561814993466570, 7.21169625723169865152885655913, 8.382724908960063077524067502734, 9.210965730279951502568192663826, 9.829214186099893558242033946108

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