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Properties

Label 2-1134-21.5-c1-0-18
Degree $2$
Conductor $1134$
Sign $0.852 + 0.522i$
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 + (−1.80 + 3.13i)5-s + (2.14 − 1.54i)7-s − 0.999i·8-s + (3.13 − 1.80i)10-s + (1.73 − 1.00i)11-s − 3.40i·13-s + (−2.63 + 0.266i)14-s + (−0.5 + 0.866i)16-s + (−3.08 − 5.34i)17-s + (0.877 + 0.506i)19-s − 3.61·20-s − 2.00·22-s + (2.62 + 1.51i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.809 + 1.40i)5-s + (0.811 − 0.584i)7-s − 0.353i·8-s + (0.991 − 0.572i)10-s + (0.523 − 0.302i)11-s − 0.945i·13-s + (−0.703 + 0.0713i)14-s + (−0.125 + 0.216i)16-s + (−0.748 − 1.29i)17-s + (0.201 + 0.116i)19-s − 0.809·20-s − 0.427·22-s + (0.546 + 0.315i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.852 + 0.522i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.852 + 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.093104641\)
\(L(\frac12)\) \(\approx\) \(1.093104641\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 + 1.54i)T \)
good5 \( 1 + (1.80 - 3.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.73 + 1.00i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.40iT - 13T^{2} \)
17 \( 1 + (3.08 + 5.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.877 - 0.506i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.62 - 1.51i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.82iT - 29T^{2} \)
31 \( 1 + (0.787 - 0.454i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.66 + 6.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.70T + 41T^{2} \)
43 \( 1 - 4.79T + 43T^{2} \)
47 \( 1 + (-1.11 + 1.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.58 + 4.37i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.49 + 7.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.7 - 7.35i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.15 - 7.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.466iT - 71T^{2} \)
73 \( 1 + (3.65 - 2.10i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.91 - 3.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.00T + 83T^{2} \)
89 \( 1 + (2.39 - 4.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
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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

−9.899304727554841556015453776485, −8.900469235804347949934106444921, −8.024901824205572998524163233324, −7.19882836409483883312997929206, −6.96688830954652411317709675118, −5.51627440903124655559016066650, −4.21628822429571350842556305124, −3.35299410018502529063826514157, −2.45836994246500995586101616295, −0.75310422547426109593166262494, 1.06979286246290929638337906263, 2.13607618781245816917332079682, 4.16002649874358824930381222615, 4.58479148139960754842649769243, 5.67299685620496983237590875840, 6.62479125147864642296195136839, 7.74511627036207283786564131731, 8.307964098755480377919945652630, 8.988681331495182881231219979817, 9.430122545573737980023521555609

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