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Properties

Label 1-205-205.204-r0-0-0
Degree $1$
Conductor $205$
Sign $1$
Analytic cond. $0.952015$
Root an. cond. $0.952015$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s + 13-s − 14-s + 16-s + 17-s − 18-s − 19-s + 21-s + 22-s − 23-s − 24-s − 26-s + 27-s + 28-s − 29-s + 31-s − 32-s − 33-s − 34-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s + 13-s − 14-s + 16-s + 17-s − 18-s − 19-s + 21-s + 22-s − 23-s − 24-s − 26-s + 27-s + 28-s − 29-s + 31-s − 32-s − 33-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(205\)    =    \(5 \cdot 41\)
Sign: $1$
Analytic conductor: \(0.952015\)
Root analytic conductor: \(0.952015\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{205} (204, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 205,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178797996\)
\(L(\frac12)\) \(\approx\) \(1.178797996\)
\(L(1)\) \(\approx\) \(1.050623227\)
\(L(1)\) \(\approx\) \(1.050623227\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
41 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.650081450523990323103382270908, −25.95487097792580062433321904997, −25.24244460715117427514097224097, −24.20338157764452992388433105640, −23.495536444298538892381093265923, −21.441769784801916958895408899801, −20.887352837830433919899423736621, −20.2491204078381543444947392479, −18.93455295003832081472092399512, −18.47203535670177198485308339104, −17.43774452291833594336007339150, −16.16176775378698397285670808550, −15.321245939053144792288746344625, −14.43421659974855760358259968221, −13.255727723864375319831488514872, −11.95919263049457276358123342659, −10.7088831302949322050595593809, −9.97311100371886289159348175316, −8.52839994915370115591979491986, −8.192765739144116330406084965851, −7.14147924049611037154518982653, −5.60893181841362294221561065449, −3.885211036652835593591011229575, −2.50253086009588512448210890372, −1.46732342903323083553918841160, 1.46732342903323083553918841160, 2.50253086009588512448210890372, 3.885211036652835593591011229575, 5.60893181841362294221561065449, 7.14147924049611037154518982653, 8.192765739144116330406084965851, 8.52839994915370115591979491986, 9.97311100371886289159348175316, 10.7088831302949322050595593809, 11.95919263049457276358123342659, 13.255727723864375319831488514872, 14.43421659974855760358259968221, 15.321245939053144792288746344625, 16.16176775378698397285670808550, 17.43774452291833594336007339150, 18.47203535670177198485308339104, 18.93455295003832081472092399512, 20.2491204078381543444947392479, 20.887352837830433919899423736621, 21.441769784801916958895408899801, 23.495536444298538892381093265923, 24.20338157764452992388433105640, 25.24244460715117427514097224097, 25.95487097792580062433321904997, 26.650081450523990323103382270908

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