Properties

Label 2-207-1.1-c5-0-35
Degree $2$
Conductor $207$
Sign $1$
Analytic cond. $33.1994$
Root an. cond. $5.76189$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.2·2-s + 72.1·4-s + 55.5·5-s + 2.50·7-s + 409.·8-s + 566.·10-s − 228.·11-s + 658.·13-s + 25.5·14-s + 1.87e3·16-s + 1.44e3·17-s − 982.·19-s + 4.00e3·20-s − 2.33e3·22-s − 529·23-s − 42.8·25-s + 6.72e3·26-s + 180.·28-s + 7.15e3·29-s − 9.25e3·31-s + 5.98e3·32-s + 1.47e4·34-s + 139.·35-s + 2.42e3·37-s − 1.00e4·38-s + 2.27e4·40-s + 4.07e3·41-s + ⋯
L(s)  = 1  + 1.80·2-s + 2.25·4-s + 0.993·5-s + 0.0193·7-s + 2.26·8-s + 1.79·10-s − 0.569·11-s + 1.08·13-s + 0.0348·14-s + 1.82·16-s + 1.21·17-s − 0.624·19-s + 2.23·20-s − 1.02·22-s − 0.208·23-s − 0.0137·25-s + 1.95·26-s + 0.0435·28-s + 1.58·29-s − 1.73·31-s + 1.03·32-s + 2.18·34-s + 0.0191·35-s + 0.290·37-s − 1.12·38-s + 2.24·40-s + 0.378·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(33.1994\)
Root analytic conductor: \(5.76189\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{207} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(7.395159475\)
\(L(\frac12)\) \(\approx\) \(7.395159475\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 + 529T \)
good2 \( 1 - 10.2T + 32T^{2} \)
5 \( 1 - 55.5T + 3.12e3T^{2} \)
7 \( 1 - 2.50T + 1.68e4T^{2} \)
11 \( 1 + 228.T + 1.61e5T^{2} \)
13 \( 1 - 658.T + 3.71e5T^{2} \)
17 \( 1 - 1.44e3T + 1.41e6T^{2} \)
19 \( 1 + 982.T + 2.47e6T^{2} \)
29 \( 1 - 7.15e3T + 2.05e7T^{2} \)
31 \( 1 + 9.25e3T + 2.86e7T^{2} \)
37 \( 1 - 2.42e3T + 6.93e7T^{2} \)
41 \( 1 - 4.07e3T + 1.15e8T^{2} \)
43 \( 1 + 1.04e4T + 1.47e8T^{2} \)
47 \( 1 + 9.35e3T + 2.29e8T^{2} \)
53 \( 1 - 3.42e4T + 4.18e8T^{2} \)
59 \( 1 - 7.26e3T + 7.14e8T^{2} \)
61 \( 1 - 2.66e4T + 8.44e8T^{2} \)
67 \( 1 - 5.34e4T + 1.35e9T^{2} \)
71 \( 1 + 2.16e4T + 1.80e9T^{2} \)
73 \( 1 + 8.28e4T + 2.07e9T^{2} \)
79 \( 1 + 2.39e4T + 3.07e9T^{2} \)
83 \( 1 + 8.11e4T + 3.93e9T^{2} \)
89 \( 1 + 1.00e5T + 5.58e9T^{2} \)
97 \( 1 - 3.61e4T + 8.58e9T^{2} \)
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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

−11.80715044539871697007724302341, −10.80859912131595228294034279293, −9.927024328009443525797820739960, −8.316848895164770491356344883305, −6.90607445899138641051915510950, −5.90791978655439757138391249423, −5.32332555503613775903217341612, −3.98427442740634116310155600764, −2.83610359431614616422093255194, −1.60851695304091276572345095134, 1.60851695304091276572345095134, 2.83610359431614616422093255194, 3.98427442740634116310155600764, 5.32332555503613775903217341612, 5.90791978655439757138391249423, 6.90607445899138641051915510950, 8.316848895164770491356344883305, 9.927024328009443525797820739960, 10.80859912131595228294034279293, 11.80715044539871697007724302341

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