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Properties

Label 2-111-111.11-c0-0-0
Degree $2$
Conductor $111$
Sign $0.957 - 0.289i$
Analytic cond. $0.0553962$
Root an. cond. $0.235364$
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  + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)12-s + (−1.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + (−0.499 − 0.866i)21-s + (0.5 + 0.866i)25-s + 0.999·27-s + (0.499 + 0.866i)28-s + 1.73i·31-s − 0.999·36-s + 37-s + (1.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)12-s + (−1.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + (−0.499 − 0.866i)21-s + (0.5 + 0.866i)25-s + 0.999·27-s + (0.499 + 0.866i)28-s + 1.73i·31-s − 0.999·36-s + 37-s + (1.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(0.0553962\)
Root analytic conductor: \(0.235364\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :0),\ 0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5278752583\)
\(L(\frac12)\) \(\approx\) \(0.5278752583\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−14.36833704050682452832322795010, −12.61525216977882003906043806501, −11.79772524475894385983329721976, −10.63744028703307335460013897015, −9.896913020367610485625389816851, −8.971057301257502278056761614457, −7.05344265564969174407287336327, −5.76006380869294265534927873326, −5.00533503922753385933189223635, −2.87499744189534508364905084632, 2.51158045381864168173649537375, 4.41656836049261226463945647116, 6.35795379858912165737410810501, 7.19538498646848317933120534875, 7.997035025629676928244448689468, 9.678190572919910280677778371520, 11.02828773337518437619578361241, 11.91496355477986837445357335560, 12.72661832174701726040763085903, 13.55016410918758895210459647472

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