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Properties

Label 1-1100-1100.423-r0-0-0
Degree $1$
Conductor $1100$
Sign $0.0798 - 0.996i$
Analytic cond. $5.10837$
Root an. cond. $5.10837$
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  + (−0.951 − 0.309i)3-s + (0.951 + 0.309i)7-s + (0.809 + 0.587i)9-s + (0.587 − 0.809i)13-s − i·17-s + (−0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (−0.951 − 0.309i)23-s + (−0.587 − 0.809i)27-s + (0.809 − 0.587i)29-s + (−0.309 + 0.951i)31-s − i·37-s + (−0.809 + 0.587i)39-s + (−0.809 + 0.587i)41-s − i·43-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)3-s + (0.951 + 0.309i)7-s + (0.809 + 0.587i)9-s + (0.587 − 0.809i)13-s − i·17-s + (−0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (−0.951 − 0.309i)23-s + (−0.587 − 0.809i)27-s + (0.809 − 0.587i)29-s + (−0.309 + 0.951i)31-s − i·37-s + (−0.809 + 0.587i)39-s + (−0.809 + 0.587i)41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0798 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0798 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.0798 - 0.996i$
Analytic conductor: \(5.10837\)
Root analytic conductor: \(5.10837\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1100,\ (0:\ ),\ 0.0798 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7653233688 - 0.7064901085i\)
\(L(\frac12)\) \(\approx\) \(0.7653233688 - 0.7064901085i\)
\(L(1)\) \(\approx\) \(0.8431164695 - 0.2047796649i\)
\(L(1)\) \(\approx\) \(0.8431164695 - 0.2047796649i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + iT \)
59 \( 1 + T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.587 - 0.809i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.56984226630787014935223992211, −21.01951441917600748877576090445, −20.23867598033312067996680547681, −19.093720053010861891121225729935, −18.37028071341051000549597512645, −17.61195917383775917403799635181, −16.948901327245252817171243970632, −16.34001658118895399151581428580, −15.38227750344360506512662984872, −14.65806862722217931001625239546, −13.78674885058754735791381862703, −12.81043478205958124141825466960, −11.923116387626423770746294032562, −11.316340219975832609935964088889, −10.550740603575516005000890651594, −9.91252442283044772123137213839, −8.69212552542503687598783137357, −7.978860035167279924201074489045, −6.82449620932184131500444925895, −6.12512919646090693974735237320, −5.25199859540603433304820154533, −4.25269423329483166195478346484, −3.80644550010351121403643863700, −1.99271878888100711412509650237, −1.21748930819477053027939492648, 0.54136524785618485965103976946, 1.67766528422268402996315763015, 2.66765154141887797503141063725, 4.13788632535191578213835726035, 4.94760650930744806041433390556, 5.67577269156032080052080641155, 6.51942662780261113032029798640, 7.48525631354430269372803844192, 8.238819972725734490193478688995, 9.172491352829424869884118622629, 10.51635137852373694221534008466, 10.795378390770597667202754460564, 11.88570861490067987370399943698, 12.26449865578772771187621012848, 13.37432179960637372815075592473, 14.01855993123335993950870633998, 15.1658732695123083030829669093, 15.79901083738605181321635915718, 16.6279097651683128487565467405, 17.60344207405931648769219336225, 17.97518187525431777325245085895, 18.62236685378459557462264306201, 19.6426532816235762211948655603, 20.55736761535583043167096663409, 21.33972852113535357566166462504

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