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Properties

Label 2-1134-21.17-c1-0-16
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: $\chi_{1134} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.561 + 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.069236583\)
\(L(\frac12)\) \(\approx\) \(1.069236583\)
\(L(\frac{3}{2})\) 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

−9.637563323056970887537956135780, −8.736027577294916117071717464848, −8.131881826799817193181718896366, −7.14672106430591206604953796377, −6.69978027060111067567806851631, −5.36867426169117214830889566264, −4.70415653471618644881833347598, −3.39037900167009256394977089205, −2.04798831911938496469652953892, −0.61992168619900710294502133729, 1.32896212676893669523191650295, 2.58648562635355701263761247313, 3.39563167708999559395016559475, 5.15680033310140731432657461582, 5.33626898583614793721120035634, 6.87036411473800357656180846728, 7.70095339032309793973745668248, 8.303830658309668690272696631875, 9.155871522867852094125334583406, 9.986550342046562106875111326995

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