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Properties

Label 2-1100-275.131-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  + (−1.08 + 0.786i)3-s + (−0.5 − 0.866i)5-s + (0.244 − 0.752i)9-s + (0.309 + 0.951i)11-s + (1.22 + 0.544i)15-s + (0.564 + 1.73i)23-s + (−0.499 + 0.866i)25-s + (−0.0864 − 0.266i)27-s + (1.58 + 1.14i)31-s + (−1.08 − 0.786i)33-s + (−0.309 + 0.951i)37-s + (−0.773 + 0.164i)45-s + (−1.61 + 1.17i)47-s + 49-s + (−0.5 + 0.363i)53-s + ⋯
L(s)  = 1  + (−1.08 + 0.786i)3-s + (−0.5 − 0.866i)5-s + (0.244 − 0.752i)9-s + (0.309 + 0.951i)11-s + (1.22 + 0.544i)15-s + (0.564 + 1.73i)23-s + (−0.499 + 0.866i)25-s + (−0.0864 − 0.266i)27-s + (1.58 + 1.14i)31-s + (−1.08 − 0.786i)33-s + (−0.309 + 0.951i)37-s + (−0.773 + 0.164i)45-s + (−1.61 + 1.17i)47-s + 49-s + (−0.5 + 0.363i)53-s + ⋯

Functional equation

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

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: $\chi_{1100} (681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :0),\ 0.146 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5929648733\)
\(L(\frac12)\) \(\approx\) \(0.5929648733\)
\(L(1)\) 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 \( 1 + (1.08 - 0.786i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (-0.564 - 1.73i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (-0.413 + 1.27i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.169 - 0.122i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \)
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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

−10.10187722017603875006450862907, −9.661615688940069576544050894224, −8.717051357436133546349404828346, −7.77122606718065139609575627008, −6.82210921522037145130234638584, −5.78525073954397145720229302754, −4.87409588129378590440542717852, −4.52876898545534418471451037015, −3.35396896572007906220366746679, −1.40619740094299035302459512857, 0.68431921546380779237457869596, 2.44119090105526815980699986467, 3.60572447141646899859445884847, 4.77365675859726681883936984807, 5.98946613136184570451864165245, 6.43081252837955954884491611285, 7.15856464406248605029335896441, 8.059575115623724879283611110687, 8.904739143174425721900092905131, 10.24544966214601491782325753490

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