Properties

Label 2-209-1.1-c1-0-5
Degree $2$
Conductor $209$
Sign $1$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.19·2-s + 3.16·3-s − 0.576·4-s + 1.08·5-s − 3.77·6-s − 1.30·7-s + 3.07·8-s + 7.00·9-s − 1.29·10-s − 11-s − 1.82·12-s + 2.53·13-s + 1.56·14-s + 3.42·15-s − 2.51·16-s − 5.43·17-s − 8.35·18-s + 19-s − 0.623·20-s − 4.14·21-s + 1.19·22-s + 3.87·23-s + 9.72·24-s − 3.82·25-s − 3.03·26-s + 12.6·27-s + 0.754·28-s + ⋯
L(s)  = 1  − 0.843·2-s + 1.82·3-s − 0.288·4-s + 0.484·5-s − 1.54·6-s − 0.494·7-s + 1.08·8-s + 2.33·9-s − 0.408·10-s − 0.301·11-s − 0.526·12-s + 0.704·13-s + 0.417·14-s + 0.883·15-s − 0.628·16-s − 1.31·17-s − 1.96·18-s + 0.229·19-s − 0.139·20-s − 0.903·21-s + 0.254·22-s + 0.807·23-s + 1.98·24-s − 0.765·25-s − 0.594·26-s + 2.43·27-s + 0.142·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $1$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.305047396\)
\(L(\frac12)\) \(\approx\) \(1.305047396\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 + 1.19T + 2T^{2} \)
3 \( 1 - 3.16T + 3T^{2} \)
5 \( 1 - 1.08T + 5T^{2} \)
7 \( 1 + 1.30T + 7T^{2} \)
13 \( 1 - 2.53T + 13T^{2} \)
17 \( 1 + 5.43T + 17T^{2} \)
23 \( 1 - 3.87T + 23T^{2} \)
29 \( 1 + 2.41T + 29T^{2} \)
31 \( 1 - 3.03T + 31T^{2} \)
37 \( 1 - 6.85T + 37T^{2} \)
41 \( 1 + 6.11T + 41T^{2} \)
43 \( 1 + 2.95T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + 0.992T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 + 5.82T + 61T^{2} \)
67 \( 1 - 8.79T + 67T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 - 5.84T + 73T^{2} \)
79 \( 1 - 17.0T + 79T^{2} \)
83 \( 1 + 7.01T + 83T^{2} \)
89 \( 1 + 9.13T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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

−12.99309940489819925338778495044, −10.98995466386863249792893386981, −9.829265547862646932767143661423, −9.378878982306308230898716684408, −8.540992584434319441278181209512, −7.83105423894568512020074972486, −6.63082079478626454569283244052, −4.58390726684160789428592755383, −3.27911967032119883950331400134, −1.83487682027436614199988471346, 1.83487682027436614199988471346, 3.27911967032119883950331400134, 4.58390726684160789428592755383, 6.63082079478626454569283244052, 7.83105423894568512020074972486, 8.540992584434319441278181209512, 9.378878982306308230898716684408, 9.829265547862646932767143661423, 10.98995466386863249792893386981, 12.99309940489819925338778495044

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