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Properties

Label 2-111-111.110-c0-0-0
Degree $2$
Conductor $111$
Sign $1$
Analytic cond. $0.0553962$
Root an. cond. $0.235364$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s − 3-s + 1.00·4-s + 1.41·5-s + 1.41·6-s + 9-s − 2.00·10-s − 1.00·12-s − 1.41·15-s − 0.999·16-s − 1.41·17-s − 1.41·18-s + 1.41·20-s + 1.41·23-s + 1.00·25-s − 27-s − 1.41·29-s + 2.00·30-s + 1.41·32-s + 2.00·34-s + 1.00·36-s − 37-s + 1.41·45-s − 2.00·46-s + 0.999·48-s − 49-s + ⋯
L(s)  = 1  − 1.41·2-s − 3-s + 1.00·4-s + 1.41·5-s + 1.41·6-s + 9-s − 2.00·10-s − 1.00·12-s − 1.41·15-s − 0.999·16-s − 1.41·17-s − 1.41·18-s + 1.41·20-s + 1.41·23-s + 1.00·25-s − 27-s − 1.41·29-s + 2.00·30-s + 1.41·32-s + 2.00·34-s + 1.00·36-s − 37-s + 1.41·45-s − 2.00·46-s + 0.999·48-s − 49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $1$
Analytic conductor: \(0.0553962\)
Root analytic conductor: \(0.235364\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (110, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2929445389\)
\(L(\frac12)\) \(\approx\) \(0.2929445389\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
37 \( 1 + T \)
good2 \( 1 + 1.41T + T^{2} \)
5 \( 1 - 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−13.63389415332974561022405771071, −12.88032595670467214222071285001, −11.26336592177830283349310279878, −10.61941457244762634938626630511, −9.627286889041242972834937905847, −8.921363424316000325266869365009, −7.23298105014718581945665391039, −6.28258528956796865322833558727, −4.94230393511775675506662434397, −1.81269192173086389162999101018, 1.81269192173086389162999101018, 4.94230393511775675506662434397, 6.28258528956796865322833558727, 7.23298105014718581945665391039, 8.921363424316000325266869365009, 9.627286889041242972834937905847, 10.61941457244762634938626630511, 11.26336592177830283349310279878, 12.88032595670467214222071285001, 13.63389415332974561022405771071

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