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Properties

Label 2-1134-21.5-c1-0-28
Degree $2$
Conductor $1134$
Sign $0.427 + 0.904i$
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 + (1.14 − 1.97i)5-s + (−2.64 + 0.117i)7-s + 0.999i·8-s + (1.97 − 1.14i)10-s + (0.946 − 0.546i)11-s − 6.82i·13-s + (−2.34 − 1.21i)14-s + (−0.5 + 0.866i)16-s + (−3.35 − 5.81i)17-s + (−2.47 − 1.43i)19-s + 2.28·20-s + 1.09·22-s + (3.38 + 1.95i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.510 − 0.883i)5-s + (−0.999 + 0.0444i)7-s + 0.353i·8-s + (0.624 − 0.360i)10-s + (0.285 − 0.164i)11-s − 1.89i·13-s + (−0.627 − 0.325i)14-s + (−0.125 + 0.216i)16-s + (−0.814 − 1.41i)17-s + (−0.568 − 0.328i)19-s + 0.510·20-s + 0.233·22-s + (0.705 + 0.407i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.427 + 0.904i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.427 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.957584990\)
\(L(\frac12)\) \(\approx\) \(1.957584990\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.117i)T \)
good5 \( 1 + (-1.14 + 1.97i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.946 + 0.546i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.82iT - 13T^{2} \)
17 \( 1 + (3.35 + 5.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.47 + 1.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.38 - 1.95i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.84iT - 29T^{2} \)
31 \( 1 + (-1.75 + 1.01i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.57 + 6.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.91T + 41T^{2} \)
43 \( 1 - 7.48T + 43T^{2} \)
47 \( 1 + (3.40 - 5.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.222 + 0.128i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.971 + 1.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.15 + 0.665i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.54 + 4.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.233iT - 71T^{2} \)
73 \( 1 + (-5.89 + 3.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.63 + 6.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.82T + 83T^{2} \)
89 \( 1 + (8.99 - 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.77iT - 97T^{2} \)
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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

−9.391741114956976601719807914533, −9.028302684793918356154006135242, −7.934590802958949530912787513047, −7.04452233097640176571405933220, −6.14496177859830440328650565907, −5.39175042272808788863447753936, −4.68990626113475858926391395883, −3.39864731168179496862731571543, −2.56644509281408758384257661154, −0.67883840113398527432746000014, 1.79919577846562315817858017438, 2.68593219017294125965889752034, 3.83753469846933674715768678343, 4.50440635666648310877920843064, 6.01250110542654796275595603340, 6.57422625102184070069950355226, 6.91886046427009082001350266027, 8.519513004645500689487979676206, 9.343113496688036433963309504731, 10.11630285420852471072735995784

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