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Properties

Label 2-1134-7.2-c1-0-23
Degree $2$
Conductor $1134$
Sign $-0.605 + 0.795i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 2.59i)7-s + 0.999·8-s − 4·13-s + (−2 − 1.73i)14-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)17-s + (−1 + 1.73i)19-s + (1.5 − 2.59i)23-s + (2.5 + 4.33i)25-s + (2 − 3.46i)26-s + (2.5 − 0.866i)28-s − 6·29-s + (−2.5 − 4.33i)31-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.188 + 0.981i)7-s + 0.353·8-s − 1.10·13-s + (−0.534 − 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.727 − 1.26i)17-s + (−0.229 + 0.397i)19-s + (0.312 − 0.541i)23-s + (0.5 + 0.866i)25-s + (0.392 − 0.679i)26-s + (0.472 − 0.163i)28-s − 1.11·29-s + (−0.449 − 0.777i)31-s + (−0.0883 − 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.260050960125288362491141173920, −8.918412189445547650265416831998, −7.77800285362632399506854219938, −7.11083353302625011988926446045, −6.22558262864935951946642542348, −5.29523508140068378484424294938, −4.63164950336747914125705986314, −3.08304247511959203085866693702, −2.00344776607527309358619456832, 0, 1.59078821258472909128980171640, 2.80346916824397796991666548518, 3.95217493127960967126349048799, 4.63805638941835510970000379343, 5.91871100919846370288569895348, 7.08995485782761256846865982345, 7.54435570008101611045531460530, 8.685503806498030119875723515906, 9.318053344123454361567767793592

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