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Properties

Label 2-1134-21.5-c1-0-19
Degree $2$
Conductor $1134$
Sign $0.561 - 0.827i$
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 + (−0.0338 + 0.0585i)5-s + (1.44 + 2.21i)7-s + 0.999i·8-s + (−0.0585 + 0.0338i)10-s + (3.40 − 1.96i)11-s − 3.84i·13-s + (0.142 + 2.64i)14-s + (−0.5 + 0.866i)16-s + (0.775 + 1.34i)17-s + (5.06 + 2.92i)19-s − 0.0676·20-s + 3.92·22-s + (−4.78 − 2.76i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.0151 + 0.0261i)5-s + (0.545 + 0.837i)7-s + 0.353i·8-s + (−0.0185 + 0.0106i)10-s + (1.02 − 0.592i)11-s − 1.06i·13-s + (0.0380 + 0.706i)14-s + (−0.125 + 0.216i)16-s + (0.188 + 0.325i)17-s + (1.16 + 0.670i)19-s − 0.0151·20-s + 0.837·22-s + (−0.998 − 0.576i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.561 - 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.624145688\)
\(L(\frac12)\) \(\approx\) \(2.624145688\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.44 - 2.21i)T \)
good5 \( 1 + (0.0338 - 0.0585i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.40 + 1.96i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.84iT - 13T^{2} \)
17 \( 1 + (-0.775 - 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.06 - 2.92i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.78 + 2.76i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.39iT - 29T^{2} \)
31 \( 1 + (1.09 - 0.632i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.35 - 7.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 1.47T + 43T^{2} \)
47 \( 1 + (-1.77 + 3.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.28 - 3.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.70 + 8.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0705 + 0.0407i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.67 - 13.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.30iT - 71T^{2} \)
73 \( 1 + (-6.12 + 3.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.42 + 5.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.86T + 83T^{2} \)
89 \( 1 + (-5.84 + 10.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.419iT - 97T^{2} \)
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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

−9.910114626157780963715319633210, −8.938942381377202262282492403120, −8.214421793830219558713236646388, −7.51757405212152721149751817990, −6.33962633779128424290524951700, −5.70472074393770401782755039396, −4.96266719780898986324337548328, −3.74125332579165863515900694163, −2.93287635579867041347313797023, −1.45831342873739133079650463774, 1.13273051938404044251932205354, 2.24826075406485869985171907409, 3.72862924992959177817902442826, 4.32102037219501967748677168415, 5.18140631314782990432409288997, 6.36676406819882137436028717292, 7.11203420078697938218635995432, 7.83385016705637477639053919940, 9.205561164706063357211941693868, 9.625524425203271177658217232322

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