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Properties

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

\(L(\frac{1}{2})\) \(\approx\) \(0.6649808684\)
\(L(\frac12)\) \(\approx\) \(0.6649808684\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
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

−10.22885078585236605198997280620, −9.393686972342165846117760456801, −8.704737437644179343292017393785, −7.984070381282567243655046343569, −6.68558424797981227747734277753, −5.61880962947641941626365815010, −4.97008673041955185435574799124, −3.64778540048786145870139323281, −2.83306801407964136421603708013, −1.80676026481348476157584741233, 0.61217288882269291288046511302, 3.02086055508473752081138004269, 3.93389485890527548095804280747, 4.89656049131498935353022522354, 5.89279361173333390665395425356, 6.79492515385218176116155095719, 7.37156073353007661457424387395, 8.002296247298560046423511421447, 9.516597118895962203902286678789, 9.650906330807742787209722592567

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