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Properties

Label 2-1134-21.5-c1-0-21
Degree $2$
Conductor $1134$
Sign $0.883 - 0.468i$
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 + (−0.483 + 0.837i)5-s + (−0.238 − 2.63i)7-s + 0.999i·8-s + (−0.837 + 0.483i)10-s + (4.82 − 2.78i)11-s + 4.35i·13-s + (1.11 − 2.40i)14-s + (−0.5 + 0.866i)16-s + (−1.97 − 3.41i)17-s + (3.86 + 2.23i)19-s − 0.967·20-s + 5.57·22-s + (2.29 + 1.32i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.216 + 0.374i)5-s + (−0.0902 − 0.995i)7-s + 0.353i·8-s + (−0.264 + 0.152i)10-s + (1.45 − 0.840i)11-s + 1.20i·13-s + (0.296 − 0.641i)14-s + (−0.125 + 0.216i)16-s + (−0.478 − 0.828i)17-s + (0.887 + 0.512i)19-s − 0.216·20-s + 1.18·22-s + (0.479 + 0.276i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.883 - 0.468i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.883 - 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.442380889\)
\(L(\frac12)\) \(\approx\) \(2.442380889\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.238 + 2.63i)T \)
good5 \( 1 + (0.483 - 0.837i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.82 + 2.78i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.35iT - 13T^{2} \)
17 \( 1 + (1.97 + 3.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.86 - 2.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.29 - 1.32i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.32iT - 29T^{2} \)
31 \( 1 + (-5.34 + 3.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.243 + 0.421i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.163T + 41T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + (4.74 - 8.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.74 + 1.00i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.836 - 1.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.47 - 2.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.72 - 4.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.64iT - 71T^{2} \)
73 \( 1 + (2.15 - 1.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.30 - 3.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.41T + 83T^{2} \)
89 \( 1 + (-2.05 + 3.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.8iT - 97T^{2} \)
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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

−9.700585745771845719203835685372, −9.156488215309293807826641716471, −8.027671408433742600905665060912, −7.09427566766492631423464341450, −6.68517996740883098081159702529, −5.75908028707357178720944906969, −4.43204304899473843431556448978, −3.90996680369670221789217909276, −2.91229592223705523157046138677, −1.19324833790616860398180551805, 1.19217785225853684595909700318, 2.50615388034312743622509231869, 3.52465637985607891702380442112, 4.59288765645516514700950021038, 5.31489471944178574426509599002, 6.31132885203022971881115857431, 7.03256775104693953639115944173, 8.305657512880835127346325944059, 8.964798274559109419531525604372, 9.764083546961556878862632244034

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