↑

Properties

Label 2-379050-1.1-c1-0-111
Degree $2$
Conductor $379050$
Sign $-1$
Analytic cond. $3026.72$
Root an. cond. $55.0157$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 5·11-s + 12-s − 13-s − 14-s + 16-s − 5·17-s − 18-s + 21-s + 5·22-s − 24-s + 26-s + 27-s + 28-s − 6·29-s − 6·31-s − 32-s − 5·33-s + 5·34-s + 36-s + 8·37-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 − 1.50·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s − 1.07·31-s − 0.176·32-s − 0.870·33-s + 0.857·34-s + 1/6·36-s + 1.31·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(379050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(3026.72\)
Root analytic conductor: \(55.0157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 379050,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
19 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−12.86327539025351, −12.35137759268148, −11.46889775135764, −11.28196652470539, −10.98999513047696, −10.28851833450412, −9.986326503887022, −9.593474206102729, −8.837459000087221, −8.761937398081384, −8.203084477477652, −7.628504128801335, −7.384804215154824, −7.023981063587217, −6.269034183541975, −5.744707039445311, −5.281444006740626, −4.771781165146504, −4.086823381918165, −3.752485809016721, −2.749444968414549, −2.623789126676686, −2.058193163063027, −1.517499456539348, −0.6293240265385397, 0, 0.6293240265385397, 1.517499456539348, 2.058193163063027, 2.623789126676686, 2.749444968414549, 3.752485809016721, 4.086823381918165, 4.771781165146504, 5.281444006740626, 5.744707039445311, 6.269034183541975, 7.023981063587217, 7.384804215154824, 7.628504128801335, 8.203084477477652, 8.761937398081384, 8.837459000087221, 9.593474206102729, 9.986326503887022, 10.28851833450412, 10.98999513047696, 11.28196652470539, 11.46889775135764, 12.35137759268148, 12.86327539025351

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