Properties

Label 2-294-1.1-c3-0-1
Degree $2$
Conductor $294$
Sign $1$
Analytic cond. $17.3465$
Root an. cond. $4.16492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s − 3.89·5-s + 6·6-s − 8·8-s + 9·9-s + 7.79·10-s − 61.3·11-s − 12·12-s + 53.6·13-s + 11.6·15-s + 16·16-s + 32.1·17-s − 18·18-s − 55.7·19-s − 15.5·20-s + 122.·22-s − 94.6·23-s + 24·24-s − 109.·25-s − 107.·26-s − 27·27-s + 138.·29-s − 23.3·30-s + 132.·31-s − 32·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.348·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.246·10-s − 1.68·11-s − 0.288·12-s + 1.14·13-s + 0.201·15-s + 0.250·16-s + 0.457·17-s − 0.235·18-s − 0.673·19-s − 0.174·20-s + 1.18·22-s − 0.857·23-s + 0.204·24-s − 0.878·25-s − 0.810·26-s − 0.192·27-s + 0.884·29-s − 0.142·30-s + 0.768·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7929441726\)
\(L(\frac12)\) \(\approx\) \(0.7929441726\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 + 3T \)
7 \( 1 \)
good5 \( 1 + 3.89T + 125T^{2} \)
11 \( 1 + 61.3T + 1.33e3T^{2} \)
13 \( 1 - 53.6T + 2.19e3T^{2} \)
17 \( 1 - 32.1T + 4.91e3T^{2} \)
19 \( 1 + 55.7T + 6.85e3T^{2} \)
23 \( 1 + 94.6T + 1.21e4T^{2} \)
29 \( 1 - 138.T + 2.43e4T^{2} \)
31 \( 1 - 132.T + 2.97e4T^{2} \)
37 \( 1 - 149.T + 5.06e4T^{2} \)
41 \( 1 - 427.T + 6.89e4T^{2} \)
43 \( 1 - 437.T + 7.95e4T^{2} \)
47 \( 1 - 57.0T + 1.03e5T^{2} \)
53 \( 1 + 263.T + 1.48e5T^{2} \)
59 \( 1 - 451.T + 2.05e5T^{2} \)
61 \( 1 - 579.T + 2.26e5T^{2} \)
67 \( 1 - 309.T + 3.00e5T^{2} \)
71 \( 1 + 1.05e3T + 3.57e5T^{2} \)
73 \( 1 - 1.19e3T + 3.89e5T^{2} \)
79 \( 1 - 1.31e3T + 4.93e5T^{2} \)
83 \( 1 - 1.19e3T + 5.71e5T^{2} \)
89 \( 1 + 233.T + 7.04e5T^{2} \)
97 \( 1 + 1.60e3T + 9.12e5T^{2} \)
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

−11.05852364847807429747357750622, −10.54657226267419943469028963146, −9.597596540950967412951135462825, −8.217287686355713759507596435813, −7.78218209660205036588708099713, −6.39613473213591813443088913526, −5.53319553785407777438267661384, −4.08187349126039454931043441182, −2.47008666819387540150876888750, −0.69779907337986548598530769885, 0.69779907337986548598530769885, 2.47008666819387540150876888750, 4.08187349126039454931043441182, 5.53319553785407777438267661384, 6.39613473213591813443088913526, 7.78218209660205036588708099713, 8.217287686355713759507596435813, 9.597596540950967412951135462825, 10.54657226267419943469028963146, 11.05852364847807429747357750622

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