Properties

Label 2-1050-1.1-c3-0-23
Degree $2$
Conductor $1050$
Sign $1$
Analytic cond. $61.9520$
Root an. cond. $7.87095$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 28·11-s − 12·12-s + 86·13-s + 14·14-s + 16·16-s + 66·17-s + 18·18-s − 48·19-s − 21·21-s + 56·22-s − 140·23-s − 24·24-s + 172·26-s − 27·27-s + 28·28-s − 34·29-s − 284·31-s + 32·32-s − 84·33-s + 132·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.767·11-s − 0.288·12-s + 1.83·13-s + 0.267·14-s + 1/4·16-s + 0.941·17-s + 0.235·18-s − 0.579·19-s − 0.218·21-s + 0.542·22-s − 1.26·23-s − 0.204·24-s + 1.29·26-s − 0.192·27-s + 0.188·28-s − 0.217·29-s − 1.64·31-s + 0.176·32-s − 0.443·33-s + 0.665·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(61.9520\)
Root analytic conductor: \(7.87095\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.418616770\)
\(L(\frac12)\) \(\approx\) \(3.418616770\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 + p T \)
5 \( 1 \)
7 \( 1 - p T \)
good11 \( 1 - 28 T + p^{3} T^{2} \)
13 \( 1 - 86 T + p^{3} T^{2} \)
17 \( 1 - 66 T + p^{3} T^{2} \)
19 \( 1 + 48 T + p^{3} T^{2} \)
23 \( 1 + 140 T + p^{3} T^{2} \)
29 \( 1 + 34 T + p^{3} T^{2} \)
31 \( 1 + 284 T + p^{3} T^{2} \)
37 \( 1 - 346 T + p^{3} T^{2} \)
41 \( 1 + 274 T + p^{3} T^{2} \)
43 \( 1 - 4 T + p^{3} T^{2} \)
47 \( 1 - 448 T + p^{3} T^{2} \)
53 \( 1 - 94 T + p^{3} T^{2} \)
59 \( 1 - 308 T + p^{3} T^{2} \)
61 \( 1 - 510 T + p^{3} T^{2} \)
67 \( 1 - 156 T + p^{3} T^{2} \)
71 \( 1 - 336 T + p^{3} T^{2} \)
73 \( 1 - 1170 T + p^{3} T^{2} \)
79 \( 1 - 16 T + p^{3} T^{2} \)
83 \( 1 + 772 T + p^{3} T^{2} \)
89 \( 1 - 1630 T + p^{3} T^{2} \)
97 \( 1 + 110 T + p^{3} T^{2} \)
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

−9.652439395970681191975563744767, −8.607208483737477087044636024104, −7.79197684079408657981087881707, −6.71971758457290754906601073718, −5.98887589602538261878889158527, −5.39132331977696056066584175261, −4.06912125586414195891354822786, −3.66741942185649126825730288876, −1.98413071173727524494695816194, −0.965979689230596066880180188151, 0.965979689230596066880180188151, 1.98413071173727524494695816194, 3.66741942185649126825730288876, 4.06912125586414195891354822786, 5.39132331977696056066584175261, 5.98887589602538261878889158527, 6.71971758457290754906601073718, 7.79197684079408657981087881707, 8.607208483737477087044636024104, 9.652439395970681191975563744767

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