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Properties

Label 2-1100-220.43-c0-0-1
Degree $2$
Conductor $1100$
Sign $0.850 - 0.525i$
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  − 2-s + 4-s + (1 − i)7-s − 8-s + i·9-s + i·11-s + (−1 + i)13-s + (−1 + i)14-s + 16-s + (1 + i)17-s − i·18-s − i·22-s + (1 − i)26-s + (1 − i)28-s − 2i·31-s − 32-s + ⋯
L(s)  = 1  − 2-s + 4-s + (1 − i)7-s − 8-s + i·9-s + i·11-s + (−1 + i)13-s + (−1 + i)14-s + 16-s + (1 + i)17-s − i·18-s − i·22-s + (1 − i)26-s + (1 − i)28-s − 2i·31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.548971\)
Root analytic conductor: \(0.740926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7339892175\)
\(L(\frac12)\) \(\approx\) \(0.7339892175\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
11 \( 1 - iT \)
good3 \( 1 - iT^{2} \)
7 \( 1 + (-1 + i)T - iT^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.03867506667980923509117520065, −9.512419985753139915283161267283, −8.295681435249154606511527064969, −7.54793500900739268082204310707, −7.37141246611777720647936643887, −6.09283913483573887504642353894, −4.88156442541400735641531739346, −4.08060610974423520257083615983, −2.33956110042994893308085790299, −1.54914759355458659143918105901, 1.02931834696667689846970028340, 2.56385856919448091187229760523, 3.34269477656623853132484367403, 5.21069098469706164380465278851, 5.68750351639636196748487568182, 6.84701278299436445848680136741, 7.70626070990460064189362363111, 8.468520736546325185773684356835, 9.049169742544896191785248058796, 9.862694777838701450234420189288

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