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Properties

Label 2-1100-5.2-c2-0-1
Degree $2$
Conductor $1100$
Sign $-0.525 + 0.850i$
Analytic cond. $29.9728$
Root an. cond. $5.47474$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.95 + 3.95i)3-s + (7.19 + 7.19i)7-s − 22.2i·9-s + 3.31·11-s + (9.00 − 9.00i)13-s + (−12.3 − 12.3i)17-s + 16.8i·19-s − 56.8·21-s + (−24.1 + 24.1i)23-s + (52.3 + 52.3i)27-s + 13.5i·29-s − 38.7·31-s + (−13.1 + 13.1i)33-s + (−17.2 − 17.2i)37-s + 71.1i·39-s + ⋯
L(s)  = 1  + (−1.31 + 1.31i)3-s + (1.02 + 1.02i)7-s − 2.47i·9-s + 0.301·11-s + (0.692 − 0.692i)13-s + (−0.727 − 0.727i)17-s + 0.887i·19-s − 2.70·21-s + (−1.04 + 1.04i)23-s + (1.93 + 1.93i)27-s + 0.467i·29-s − 1.25·31-s + (−0.397 + 0.397i)33-s + (−0.465 − 0.465i)37-s + 1.82i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(29.9728\)
Root analytic conductor: \(5.47474\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2799144120\)
\(L(\frac12)\) \(\approx\) \(0.2799144120\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 - 3.31T \)
good3 \( 1 + (3.95 - 3.95i)T - 9iT^{2} \)
7 \( 1 + (-7.19 - 7.19i)T + 49iT^{2} \)
13 \( 1 + (-9.00 + 9.00i)T - 169iT^{2} \)
17 \( 1 + (12.3 + 12.3i)T + 289iT^{2} \)
19 \( 1 - 16.8iT - 361T^{2} \)
23 \( 1 + (24.1 - 24.1i)T - 529iT^{2} \)
29 \( 1 - 13.5iT - 841T^{2} \)
31 \( 1 + 38.7T + 961T^{2} \)
37 \( 1 + (17.2 + 17.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 18.8T + 1.68e3T^{2} \)
43 \( 1 + (46.8 - 46.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (24.1 + 24.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (-42.8 + 42.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 5.26iT - 3.48e3T^{2} \)
61 \( 1 + 95.5T + 3.72e3T^{2} \)
67 \( 1 + (41.5 + 41.5i)T + 4.48e3iT^{2} \)
71 \( 1 + 9.80T + 5.04e3T^{2} \)
73 \( 1 + (57.7 - 57.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 42.6iT - 6.24e3T^{2} \)
83 \( 1 + (-82.6 + 82.6i)T - 6.88e3iT^{2} \)
89 \( 1 - 24.6iT - 7.92e3T^{2} \)
97 \( 1 + (-19.7 - 19.7i)T + 9.40e3iT^{2} \)
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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

−10.35862418460868999204469199215, −9.445281024439143300677410956634, −8.837586876996925290432437224998, −7.83984969126822379016246133552, −6.49269758176382080223101127565, −5.60165758932172576288235534860, −5.28687993920050102570620201153, −4.27854245139119693411322812594, −3.37663492948700986553526495100, −1.62074193365030850908237871875, 0.10708507119910572451662268207, 1.31400566261518904807229930977, 2.03812960565016157906641371641, 4.12043973978028234041245470473, 4.81107560553549631329040014232, 5.94597331469915917297157311831, 6.61134252818139300427643883754, 7.26939443277096406680534135679, 8.043137026024447710704940551827, 8.875657708123399805727241624104

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