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Properties

Label 2-1100-275.219-c0-0-0
Degree $2$
Conductor $1100$
Sign $0.146 - 0.989i$
Analytic cond. $0.548971$
Root an. cond. $0.740926$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.873 + 1.20i)3-s + (0.5 + 0.866i)5-s + (−0.373 + 1.14i)9-s + (−0.309 − 0.951i)11-s + (−0.604 + 1.35i)15-s + (0.773 − 0.251i)23-s + (−0.499 + 0.866i)25-s + (−0.295 + 0.0960i)27-s + (−1.58 − 1.14i)31-s + (0.873 − 1.20i)33-s + (−1.64 − 0.535i)37-s + (−1.18 + 0.251i)45-s − 49-s + (1.11 + 1.53i)53-s + (0.669 − 0.743i)55-s + ⋯
L(s)  = 1  + (0.873 + 1.20i)3-s + (0.5 + 0.866i)5-s + (−0.373 + 1.14i)9-s + (−0.309 − 0.951i)11-s + (−0.604 + 1.35i)15-s + (0.773 − 0.251i)23-s + (−0.499 + 0.866i)25-s + (−0.295 + 0.0960i)27-s + (−1.58 − 1.14i)31-s + (0.873 − 1.20i)33-s + (−1.64 − 0.535i)37-s + (−1.18 + 0.251i)45-s − 49-s + (1.11 + 1.53i)53-s + (0.669 − 0.743i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.146 - 0.989i$
Analytic conductor: \(0.548971\)
Root analytic conductor: \(0.740926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :0),\ 0.146 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.438657381\)
\(L(\frac12)\) \(\approx\) \(1.438657381\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.873 - 1.20i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.773 + 0.251i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (1.58 + 1.14i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (1.64 + 0.535i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.413 + 1.27i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-1.16 + 1.60i)T + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (1.08 - 0.786i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.0646 - 0.198i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.244 - 0.336i)T + (-0.309 + 0.951i)T^{2} \)
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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

−10.24197594571595717584005867703, −9.367227912628026807586660675391, −8.874659297894971893372762752528, −7.924793122821666221360233492859, −6.96508600781463108751489802529, −5.88032093367915281448843133370, −5.05362137638548325051158275845, −3.78533184985689360915196138135, −3.21485288943417913671195187452, −2.22164132887692981900178327870, 1.43720025654986251813518532455, 2.17867466567828579123617630787, 3.39500040508258854286872999824, 4.78663062930310487871220596758, 5.58628650757131943795466055669, 6.90395988957023058371163930938, 7.26426483510555902009738322112, 8.377828544860265031975220234809, 8.790268675154382028105064005485, 9.667455645908098195051830171076

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