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Properties

Label 2-1134-63.5-c1-0-15
Degree $2$
Conductor $1134$
Sign $0.182 + 0.983i$
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  − i·2-s − 4-s + (−1.22 − 2.12i)5-s + (2.62 − 0.358i)7-s + i·8-s + (−2.12 + 1.22i)10-s + (3.67 + 2.12i)11-s + (3.62 + 2.09i)13-s + (−0.358 − 2.62i)14-s + 16-s + (−1.22 − 2.12i)17-s + (4.24 + 2.44i)19-s + (1.22 + 2.12i)20-s + (2.12 − 3.67i)22-s + (−5.19 + 3i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.547 − 0.948i)5-s + (0.990 − 0.135i)7-s + 0.353i·8-s + (−0.670 + 0.387i)10-s + (1.10 + 0.639i)11-s + (1.00 + 0.579i)13-s + (−0.0958 − 0.700i)14-s + 0.250·16-s + (−0.297 − 0.514i)17-s + (0.973 + 0.561i)19-s + (0.273 + 0.474i)20-s + (0.452 − 0.783i)22-s + (−1.08 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.182 + 0.983i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.182 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.744589960\)
\(L(\frac12)\) \(\approx\) \(1.744589960\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (-2.62 + 0.358i)T \)
good5 \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.67 - 2.12i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.62 - 2.09i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.24 - 2.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.87 + 5.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.61iT - 31T^{2} \)
37 \( 1 + (-1.62 + 2.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.22 + 2.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.94T + 47T^{2} \)
53 \( 1 + (-2.15 + 1.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.44T + 59T^{2} \)
61 \( 1 - 0.717iT - 61T^{2} \)
67 \( 1 + 3.48T + 67T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (13.2 - 7.64i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.24T + 79T^{2} \)
83 \( 1 + (-2.74 - 4.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.87 + 15.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.74 + 3.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.609320006720465231776192281942, −8.797368642322222274675612726605, −8.268437095208625338282583301688, −7.35527872169449936870460906636, −6.17369462702358746406174881964, −4.95525597123336026128138993930, −4.35181920288935479426975811294, −3.59216824933628714410565689020, −1.86454365711732429432244125663, −1.04212164399707012814132861656, 1.19693929740338588308961457731, 2.99814123435752513156012241553, 3.90822630776541845787388495475, 4.85582945185529750376317228279, 6.07682907449036740123607053311, 6.54450787831757902538090400586, 7.58845340620133918063153971843, 8.275231359462432122370677456113, 8.836675729762017274527227630906, 10.00531914463315031674899018377

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