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Properties

Label 2-1140-19.11-c1-0-0
Degree $2$
Conductor $1140$
Sign $-0.0977 - 0.995i$
Analytic cond. $9.10294$
Root an. cond. $3.01710$
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  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s − 2·7-s + (−0.499 − 0.866i)9-s − 2·11-s + (0.499 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 4.33i)19-s + (−1 + 1.73i)21-s + (4 + 6.92i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (3 + 5.19i)29-s − 3·31-s + (−1 + 1.73i)33-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.223 + 0.387i)5-s − 0.755·7-s + (−0.166 − 0.288i)9-s − 0.603·11-s + (0.129 + 0.223i)15-s + (−0.121 + 0.210i)17-s + (−0.114 + 0.993i)19-s + (−0.218 + 0.377i)21-s + (0.834 + 1.44i)23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s + (0.557 + 0.964i)29-s − 0.538·31-s + (−0.174 + 0.301i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.0977 - 0.995i$
Analytic conductor: \(9.10294\)
Root analytic conductor: \(3.01710\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :1/2),\ -0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8997763025\)
\(L(\frac12)\) \(\approx\) \(0.8997763025\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 4.33i)T \)
good7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (2 - 3.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6 - 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.5 + 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3 - 5.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4 - 6.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11T + 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.936633683336715192133654461454, −9.273543597525241943558225486854, −8.220413440744960637054566700392, −7.59494672793727622316772578214, −6.73094442762432761522906994012, −6.00564050687361173268780622435, −4.92770262612755435619333366564, −3.53830546618600769444687459202, −2.94951358973155154654716209911, −1.53749101803381352246038596213, 0.37173987528240481263761181968, 2.39866234635984487171182468256, 3.28166449189888195326264399220, 4.42771129142824449628815398124, 5.10247035472623644825125004760, 6.25729620937100098520054816486, 7.10764730579910796314033638024, 8.107650332960755336386650748805, 8.879933277381609031802718611186, 9.490569466185323981936903230696

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