Properties

Label 2-294-1.1-c1-0-2
Degree $2$
Conductor $294$
Sign $1$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 5·11-s − 12-s − 15-s + 16-s − 4·17-s + 18-s + 8·19-s + 20-s + 5·22-s − 4·23-s − 24-s − 4·25-s − 27-s − 5·29-s − 30-s + 3·31-s + 32-s − 5·33-s − 4·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 1.06·22-s − 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.192·27-s − 0.928·29-s − 0.182·30-s + 0.538·31-s + 0.176·32-s − 0.870·33-s − 0.685·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.828000428\)
\(L(\frac12)\) \(\approx\) \(1.828000428\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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

−11.74523480292824260887423342382, −11.22224988296755911606023262749, −9.923651152419313062412512140731, −9.180591929632892398823763196977, −7.62779471882474450110803429137, −6.55776248876886813154572094683, −5.83426012328410247998142577853, −4.67074113127017439545556449138, −3.52909847925109475796916640798, −1.68608286850308687773629845560, 1.68608286850308687773629845560, 3.52909847925109475796916640798, 4.67074113127017439545556449138, 5.83426012328410247998142577853, 6.55776248876886813154572094683, 7.62779471882474450110803429137, 9.180591929632892398823763196977, 9.923651152419313062412512140731, 11.22224988296755911606023262749, 11.74523480292824260887423342382

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