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Properties

Label 2-1100-55.32-c1-0-3
Degree $2$
Conductor $1100$
Sign $0.117 - 0.993i$
Analytic cond. $8.78354$
Root an. cond. $2.96370$
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  + (0.323 − 0.323i)3-s + (0.901 − 0.901i)7-s + 2.79i·9-s + (−1.79 + 2.79i)11-s + (2.51 + 2.51i)13-s + (−5.22 + 5.22i)17-s − 5·19-s − 0.582i·21-s + (2.83 − 2.83i)23-s + (1.87 + 1.87i)27-s − 8.37·29-s + 4.58·31-s + (0.323 + 1.48i)33-s + (4.96 + 4.96i)37-s + 1.62·39-s + ⋯
L(s)  = 1  + (0.186 − 0.186i)3-s + (0.340 − 0.340i)7-s + 0.930i·9-s + (−0.540 + 0.841i)11-s + (0.698 + 0.698i)13-s + (−1.26 + 1.26i)17-s − 1.14·19-s − 0.127i·21-s + (0.592 − 0.592i)23-s + (0.360 + 0.360i)27-s − 1.55·29-s + 0.823·31-s + (0.0562 + 0.257i)33-s + (0.816 + 0.816i)37-s + 0.260·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.117 - 0.993i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ 0.117 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.373290110\)
\(L(\frac12)\) \(\approx\) \(1.373290110\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 + (1.79 - 2.79i)T \)
good3 \( 1 + (-0.323 + 0.323i)T - 3iT^{2} \)
7 \( 1 + (-0.901 + 0.901i)T - 7iT^{2} \)
13 \( 1 + (-2.51 - 2.51i)T + 13iT^{2} \)
17 \( 1 + (5.22 - 5.22i)T - 17iT^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-2.83 + 2.83i)T - 23iT^{2} \)
29 \( 1 + 8.37T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 + (-4.96 - 4.96i)T + 37iT^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 + (-6.83 - 6.83i)T + 43iT^{2} \)
47 \( 1 + (0.578 + 0.578i)T + 47iT^{2} \)
53 \( 1 + (-7.09 + 7.09i)T - 53iT^{2} \)
59 \( 1 - 8.58iT - 59T^{2} \)
61 \( 1 - 7.79iT - 61T^{2} \)
67 \( 1 + (6.19 + 6.19i)T + 67iT^{2} \)
71 \( 1 + 0.417T + 71T^{2} \)
73 \( 1 + (-0.901 - 0.901i)T + 73iT^{2} \)
79 \( 1 + 7.20T + 79T^{2} \)
83 \( 1 + (-5.22 - 5.22i)T + 83iT^{2} \)
89 \( 1 + 4.79iT - 89T^{2} \)
97 \( 1 + (-3.35 - 3.35i)T + 97iT^{2} \)
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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

−10.24891154124947401850312772616, −9.026237277565972267745147069174, −8.407500775644308947417947073649, −7.61300378212449512328718707479, −6.77421484596655795380562266592, −5.89666699301639009314891451740, −4.57730306573068018537114096775, −4.18396287574279451974598047570, −2.48302379069374012227227053807, −1.69271147179541110006609244539, 0.58082406430607239866596985523, 2.35521728604292314290350220223, 3.34656495077392585144628936113, 4.34674496586969934096368602194, 5.46700004930107318478413629937, 6.19070731289006391600214773565, 7.17636293198180912854310485021, 8.190136224159582784529525700727, 8.912791760668481940371282434589, 9.422921238612726626224994526585

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