Properties

Label 2-3850-1.1-c1-0-9
Degree $2$
Conductor $3850$
Sign $1$
Analytic cond. $30.7424$
Root an. cond. $5.54458$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s − 11-s − 2·12-s − 2·13-s − 14-s + 16-s − 2·17-s + 18-s + 6·19-s + 2·21-s − 22-s − 6·23-s − 2·24-s − 2·26-s + 4·27-s − 28-s + 4·29-s + 32-s + 2·33-s − 2·34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.37·19-s + 0.436·21-s − 0.213·22-s − 1.25·23-s − 0.408·24-s − 0.392·26-s + 0.769·27-s − 0.188·28-s + 0.742·29-s + 0.176·32-s + 0.348·33-s − 0.342·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3850\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(30.7424\)
Root analytic conductor: \(5.54458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.458367802\)
\(L(\frac12)\) \(\approx\) \(1.458367802\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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

−8.364236767343812566058459791149, −7.47061829047941771853851611165, −6.78841724535451684686699565182, −6.15479195176174877530934120052, −5.37951608966004812570454158780, −4.98991370688772168201208158046, −4.00243555969074118936455684261, −3.09000934358695030931506608380, −2.09189918687403940235371222096, −0.64470210747492339700988718720, 0.64470210747492339700988718720, 2.09189918687403940235371222096, 3.09000934358695030931506608380, 4.00243555969074118936455684261, 4.98991370688772168201208158046, 5.37951608966004812570454158780, 6.15479195176174877530934120052, 6.78841724535451684686699565182, 7.47061829047941771853851611165, 8.364236767343812566058459791149

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