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Properties

Label 2-1100-275.131-c0-0-1
Degree $2$
Conductor $1100$
Sign $0.783 + 0.621i$
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  + (1.58 − 1.14i)3-s + (−0.5 + 0.866i)5-s + (0.873 − 2.68i)9-s + (0.309 + 0.951i)11-s + (0.204 + 1.94i)15-s + (−0.0646 − 0.198i)23-s + (−0.499 − 0.866i)25-s + (−1.10 − 3.39i)27-s + (−1.08 − 0.786i)31-s + (1.58 + 1.14i)33-s + (−0.309 + 0.951i)37-s + (1.89 + 2.10i)45-s + (−1.61 + 1.17i)47-s + 49-s + (−0.5 + 0.363i)53-s + ⋯
L(s)  = 1  + (1.58 − 1.14i)3-s + (−0.5 + 0.866i)5-s + (0.873 − 2.68i)9-s + (0.309 + 0.951i)11-s + (0.204 + 1.94i)15-s + (−0.0646 − 0.198i)23-s + (−0.499 − 0.866i)25-s + (−1.10 − 3.39i)27-s + (−1.08 − 0.786i)31-s + (1.58 + 1.14i)33-s + (−0.309 + 0.951i)37-s + (1.89 + 2.10i)45-s + (−1.61 + 1.17i)47-s + 49-s + (−0.5 + 0.363i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(0.548971\)
Root analytic conductor: \(0.740926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :0),\ 0.783 + 0.621i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.590766545\)
\(L(\frac12)\) \(\approx\) \(1.590766545\)
\(L(1)\) 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 \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (0.0646 + 0.198i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.604 - 1.86i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (1.47 + 1.07i)T + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (-0.564 - 1.73i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (1.08 - 0.786i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.666544596733345526210689076567, −9.098504520237145983917082854752, −8.046974264109641573019447107835, −7.59666778860703597920945463014, −6.88489128098119817305796182381, −6.22628080896827549335875127382, −4.36494773804077246614441632900, −3.45896928822767482876833772842, −2.61947674019799653979113445956, −1.65697081987161917479303215584, 1.85940262462631432300436045114, 3.28486306474004199254339630574, 3.77294447858307119124118709170, 4.73888725886720569479095735400, 5.52558717328011938729801539270, 7.17117091453817245846020143814, 8.069996868194573985700488592692, 8.627373941793174635642108307407, 9.140264297217839502726045868682, 9.878541706414430796694513460732

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