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Properties

Label 2-207-207.22-c0-0-1
Degree $2$
Conductor $207$
Sign $1$
Analytic cond. $0.103306$
Root an. cond. $0.321413$
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  + (−0.173 − 0.300i)2-s + (0.173 + 0.984i)3-s + (0.439 − 0.761i)4-s + (0.266 − 0.223i)6-s − 0.652·8-s + (−0.939 + 0.342i)9-s + (0.826 + 0.300i)12-s + (−0.173 + 0.300i)13-s + (−0.326 − 0.565i)16-s + (0.266 + 0.223i)18-s + (−0.5 + 0.866i)23-s + (−0.113 − 0.642i)24-s + (−0.5 − 0.866i)25-s + 0.120·26-s + (−0.5 − 0.866i)27-s + ⋯
L(s)  = 1  + (−0.173 − 0.300i)2-s + (0.173 + 0.984i)3-s + (0.439 − 0.761i)4-s + (0.266 − 0.223i)6-s − 0.652·8-s + (−0.939 + 0.342i)9-s + (0.826 + 0.300i)12-s + (−0.173 + 0.300i)13-s + (−0.326 − 0.565i)16-s + (0.266 + 0.223i)18-s + (−0.5 + 0.866i)23-s + (−0.113 − 0.642i)24-s + (−0.5 − 0.866i)25-s + 0.120·26-s + (−0.5 − 0.866i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(0.103306\)
Root analytic conductor: \(0.321413\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7008354182\)
\(L(\frac12)\) \(\approx\) \(0.7008354182\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.53T + T^{2} \)
73 \( 1 - 1.53T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−12.34844435424194060114802705425, −11.36075739623383129088344898563, −10.66642949741204061087781368856, −9.727935506054881373009438373879, −9.112661889037861863423111246919, −7.72298538475102849771014263239, −6.21378919839779505524566768705, −5.25379888439900349615614016668, −3.87015542094150353159278466892, −2.31067133706731657237815546333, 2.25393059789021078712292873229, 3.61010769354837001529069428782, 5.64414085076253983273149391896, 6.75111261241441636550738520954, 7.57545690836230900189433925733, 8.358133259469985566191562104336, 9.387291083643073315123371721123, 10.99259862295239296513281005224, 11.80720584062588877339614730532, 12.71531196864735011227422495182

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