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Properties

Label 2-1134-21.20-c1-0-7
Degree $2$
Conductor $1134$
Sign $-0.202 - 0.979i$
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 − 3.79·5-s + (2.59 − 0.536i)7-s − i·8-s − 3.79i·10-s + 1.61i·11-s − 5.11i·13-s + (0.536 + 2.59i)14-s + 16-s + 2.33·17-s + 4.24i·19-s + 3.79·20-s − 1.61·22-s + 2.62i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.69·5-s + (0.979 − 0.202i)7-s − 0.353i·8-s − 1.20i·10-s + 0.486i·11-s − 1.41i·13-s + (0.143 + 0.692i)14-s + 0.250·16-s + 0.565·17-s + 0.974i·19-s + 0.849·20-s − 0.344·22-s + 0.548i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.202 - 0.979i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.050217783\)
\(L(\frac12)\) \(\approx\) \(1.050217783\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 0.536i)T \)
good5 \( 1 + 3.79T + 5T^{2} \)
11 \( 1 - 1.61iT - 11T^{2} \)
13 \( 1 + 5.11iT - 13T^{2} \)
17 \( 1 - 2.33T + 17T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 - 2.62iT - 23T^{2} \)
29 \( 1 - 4.01iT - 29T^{2} \)
31 \( 1 - 7.98iT - 31T^{2} \)
37 \( 1 + 5.74T + 37T^{2} \)
41 \( 1 - 8.51T + 41T^{2} \)
43 \( 1 - 9.25T + 43T^{2} \)
47 \( 1 + 9.69T + 47T^{2} \)
53 \( 1 + 2.19iT - 53T^{2} \)
59 \( 1 + 1.46T + 59T^{2} \)
61 \( 1 - 14.4iT - 61T^{2} \)
67 \( 1 + 5.78T + 67T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 - 7.65iT - 73T^{2} \)
79 \( 1 + 1.64T + 79T^{2} \)
83 \( 1 - 3.02T + 83T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−10.18301728567254102528485683327, −8.810183676135988228392803378309, −8.166147492467712662818207610303, −7.58957180245854067662860174660, −7.14778608043309462218922546227, −5.68011633509555203008909589152, −4.92696841452649303540361880966, −4.02579131482123047584582448251, −3.20345302912209861016352375582, −1.11157180246914796440973893257, 0.58181891852603560515120685357, 2.14069027625292992509629911564, 3.40175043610569181816541128107, 4.33292470663216775337241165148, 4.78200028633101794184566921225, 6.21574371005572351303129776646, 7.45260973309649866301369069649, 7.957586994612101359137186666141, 8.797639424761784745229896852284, 9.447299508343835009775297985296

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