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Properties

Label 2-111-111.56-c1-0-4
Degree $2$
Conductor $111$
Sign $0.914 - 0.404i$
Analytic cond. $0.886339$
Root an. cond. $0.941456$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.11 + 1.32i)3-s + (1.96 − 0.347i)4-s + (−4.26 − 1.55i)7-s + (−0.520 + 2.95i)9-s + (2.65 + 2.22i)12-s + (1.33 − 1.91i)13-s + (3.75 − 1.36i)16-s + (−5.39 − 0.472i)19-s + (−2.68 − 7.39i)21-s + (−3.21 + 3.83i)25-s + (−4.5 + 2.59i)27-s + (−8.94 − 1.57i)28-s + (7.85 − 7.85i)31-s + 6i·36-s + (0.5 + 6.06i)37-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)3-s + (0.984 − 0.173i)4-s + (−1.61 − 0.586i)7-s + (−0.173 + 0.984i)9-s + (0.766 + 0.642i)12-s + (0.371 − 0.530i)13-s + (0.939 − 0.342i)16-s + (−1.23 − 0.108i)19-s + (−0.586 − 1.61i)21-s + (−0.642 + 0.766i)25-s + (−0.866 + 0.499i)27-s + (−1.69 − 0.298i)28-s + (1.41 − 1.41i)31-s + i·36-s + (0.0821 + 0.996i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :1/2),\ 0.914 - 0.404i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26783 + 0.268218i\)
\(L(\frac12)\) \(\approx\) \(1.26783 + 0.268218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 - 1.32i)T \)
37 \( 1 + (-0.5 - 6.06i)T \)
good2 \( 1 + (-1.96 + 0.347i)T^{2} \)
5 \( 1 + (3.21 - 3.83i)T^{2} \)
7 \( 1 + (4.26 + 1.55i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.33 + 1.91i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (-5.81 + 15.9i)T^{2} \)
19 \( 1 + (5.39 + 0.472i)T + (18.7 + 3.29i)T^{2} \)
23 \( 1 + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-7.85 + 7.85i)T - 31iT^{2} \)
41 \( 1 + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (4.99 + 4.99i)T + 43iT^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-37.9 - 45.1i)T^{2} \)
61 \( 1 + (-12.1 - 8.48i)T + (20.8 + 57.3i)T^{2} \)
67 \( 1 + (4.95 - 13.6i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-12.3 + 69.9i)T^{2} \)
73 \( 1 - 1.24iT - 73T^{2} \)
79 \( 1 + (-7.14 - 15.3i)T + (-50.7 + 60.5i)T^{2} \)
83 \( 1 + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (57.2 + 68.1i)T^{2} \)
97 \( 1 + (-3.92 + 14.6i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.64493083244295565821073220790, −12.90354742409520850180595004131, −11.42025954446056207986588693956, −10.28711904413150066502568560531, −9.811372101648210865329672807473, −8.333181422571698296432951599895, −7.00715989402632840904281874581, −5.90451136210303021928883235317, −3.90470699386257629160462430551, −2.73569638412466452341540817502, 2.29786363206885005684567925358, 3.48431701449349427265678922250, 6.31189247099727547053051491671, 6.60247695848260126833264574818, 8.082311346049422732809309861617, 9.167847185961505258293335212465, 10.37159604269299434485401429314, 11.87857851072515526478839460377, 12.52094774836111971434168721967, 13.37951510977483260158278122629

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