Properties

Label 2-207-1.1-c5-0-19
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9.34·2-s + 55.4·4-s − 70.5·5-s − 89.7·7-s − 218.·8-s + 659.·10-s + 436.·11-s − 258.·13-s + 838.·14-s + 272.·16-s − 591.·17-s + 1.66e3·19-s − 3.90e3·20-s − 4.08e3·22-s + 529·23-s + 1.84e3·25-s + 2.42e3·26-s − 4.97e3·28-s + 5.12e3·29-s + 1.87e3·31-s + 4.45e3·32-s + 5.53e3·34-s + 6.32e3·35-s + 884.·37-s − 1.55e4·38-s + 1.54e4·40-s + 1.84e4·41-s + ⋯
L(s)  = 1  − 1.65·2-s + 1.73·4-s − 1.26·5-s − 0.692·7-s − 1.20·8-s + 2.08·10-s + 1.08·11-s − 0.424·13-s + 1.14·14-s + 0.266·16-s − 0.496·17-s + 1.05·19-s − 2.18·20-s − 1.79·22-s + 0.208·23-s + 0.591·25-s + 0.702·26-s − 1.19·28-s + 1.13·29-s + 0.349·31-s + 0.768·32-s + 0.820·34-s + 0.872·35-s + 0.106·37-s − 1.74·38-s + 1.52·40-s + 1.71·41-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: \(1\)
Selberg data: \((2,\ 207,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 9.34T + 32T^{2} \)
5 \( 1 + 70.5T + 3.12e3T^{2} \)
7 \( 1 + 89.7T + 1.68e4T^{2} \)
11 \( 1 - 436.T + 1.61e5T^{2} \)
13 \( 1 + 258.T + 3.71e5T^{2} \)
17 \( 1 + 591.T + 1.41e6T^{2} \)
19 \( 1 - 1.66e3T + 2.47e6T^{2} \)
29 \( 1 - 5.12e3T + 2.05e7T^{2} \)
31 \( 1 - 1.87e3T + 2.86e7T^{2} \)
37 \( 1 - 884.T + 6.93e7T^{2} \)
41 \( 1 - 1.84e4T + 1.15e8T^{2} \)
43 \( 1 - 1.23e4T + 1.47e8T^{2} \)
47 \( 1 + 2.38e4T + 2.29e8T^{2} \)
53 \( 1 - 1.45e4T + 4.18e8T^{2} \)
59 \( 1 + 4.77e4T + 7.14e8T^{2} \)
61 \( 1 + 1.53e4T + 8.44e8T^{2} \)
67 \( 1 + 4.11e4T + 1.35e9T^{2} \)
71 \( 1 - 4.56e4T + 1.80e9T^{2} \)
73 \( 1 + 7.89e3T + 2.07e9T^{2} \)
79 \( 1 - 7.14e4T + 3.07e9T^{2} \)
83 \( 1 + 1.12e5T + 3.93e9T^{2} \)
89 \( 1 + 1.37e5T + 5.58e9T^{2} \)
97 \( 1 - 4.16e4T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.95117598892765534978867273016, −9.756019194797195128235835623756, −9.118169119307383881838617924732, −8.080976790296397892532181294580, −7.27959718696764679201917651309, −6.40848084673431174650897636036, −4.33688325466885155897023616713, −2.93685159139078593387133438247, −1.11475956789843685528933425306, 0, 1.11475956789843685528933425306, 2.93685159139078593387133438247, 4.33688325466885155897023616713, 6.40848084673431174650897636036, 7.27959718696764679201917651309, 8.080976790296397892532181294580, 9.118169119307383881838617924732, 9.756019194797195128235835623756, 10.95117598892765534978867273016

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