Properties

Label 2-3150-1.1-c1-0-24
Degree $2$
Conductor $3150$
Sign $1$
Analytic cond. $25.1528$
Root an. cond. $5.01526$
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 + 4-s + 7-s + 8-s + 6·11-s − 2·13-s + 14-s + 16-s − 2·17-s + 4·19-s + 6·22-s − 4·23-s − 2·26-s + 28-s + 2·29-s − 2·31-s + 32-s − 2·34-s + 10·37-s + 4·38-s + 6·41-s + 2·43-s + 6·44-s − 4·46-s + 2·47-s + 49-s − 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.80·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 1.27·22-s − 0.834·23-s − 0.392·26-s + 0.188·28-s + 0.371·29-s − 0.359·31-s + 0.176·32-s − 0.342·34-s + 1.64·37-s + 0.648·38-s + 0.937·41-s + 0.304·43-s + 0.904·44-s − 0.589·46-s + 0.291·47-s + 1/7·49-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(25.1528\)
Root analytic conductor: \(5.01526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.484012067\)
\(L(\frac12)\) \(\approx\) \(3.484012067\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−8.750248681178999332260175128359, −7.70535913170137205376297116438, −7.15871738513877227188953947019, −6.25456367993631922501934529820, −5.74658732419974295149136229246, −4.57237140963321973928767492280, −4.18629878661942399335793358254, −3.19072037140965421558183680805, −2.13374395341447365479458704361, −1.09521871351533512349503789546, 1.09521871351533512349503789546, 2.13374395341447365479458704361, 3.19072037140965421558183680805, 4.18629878661942399335793358254, 4.57237140963321973928767492280, 5.74658732419974295149136229246, 6.25456367993631922501934529820, 7.15871738513877227188953947019, 7.70535913170137205376297116438, 8.750248681178999332260175128359

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