Properties

Label 2-7448-1.1-c1-0-16
Degree $2$
Conductor $7448$
Sign $1$
Analytic cond. $59.4725$
Root an. cond. $7.71184$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.436·3-s − 0.299·5-s − 2.80·9-s − 5.18·11-s − 1.26·13-s + 0.130·15-s + 1.95·17-s + 19-s − 1.49·23-s − 4.91·25-s + 2.53·27-s − 0.530·29-s + 1.94·31-s + 2.26·33-s + 4.19·37-s + 0.550·39-s − 2·41-s + 5.51·43-s + 0.841·45-s − 8.54·47-s − 0.855·51-s − 11.7·53-s + 1.55·55-s − 0.436·57-s − 12.8·59-s − 5.93·61-s + 0.378·65-s + ⋯
L(s)  = 1  − 0.252·3-s − 0.134·5-s − 0.936·9-s − 1.56·11-s − 0.349·13-s + 0.0337·15-s + 0.475·17-s + 0.229·19-s − 0.310·23-s − 0.982·25-s + 0.488·27-s − 0.0984·29-s + 0.349·31-s + 0.394·33-s + 0.689·37-s + 0.0882·39-s − 0.312·41-s + 0.841·43-s + 0.125·45-s − 1.24·47-s − 0.119·51-s − 1.61·53-s + 0.209·55-s − 0.0578·57-s − 1.66·59-s − 0.760·61-s + 0.0468·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7448\)    =    \(2^{3} \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(59.4725\)
Root analytic conductor: \(7.71184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7448} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7448,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7758136661\)
\(L(\frac12)\) \(\approx\) \(0.7758136661\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 0.436T + 3T^{2} \)
5 \( 1 + 0.299T + 5T^{2} \)
11 \( 1 + 5.18T + 11T^{2} \)
13 \( 1 + 1.26T + 13T^{2} \)
17 \( 1 - 1.95T + 17T^{2} \)
23 \( 1 + 1.49T + 23T^{2} \)
29 \( 1 + 0.530T + 29T^{2} \)
31 \( 1 - 1.94T + 31T^{2} \)
37 \( 1 - 4.19T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 + 8.54T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 + 5.93T + 61T^{2} \)
67 \( 1 - 5.50T + 67T^{2} \)
71 \( 1 - 2.28T + 71T^{2} \)
73 \( 1 + 4.69T + 73T^{2} \)
79 \( 1 - 5.69T + 79T^{2} \)
83 \( 1 - 1.46T + 83T^{2} \)
89 \( 1 + 3.85T + 89T^{2} \)
97 \( 1 + 9.69T + 97T^{2} \)
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

−7.903747642917268072119113425258, −7.42105783338354734660765287895, −6.30144389618192892422436927345, −5.81750765150096166009137204014, −5.10922599610315259708252606383, −4.52453576156045531978634360522, −3.33733236362799977187400306883, −2.80398561467016136394838585001, −1.88039250670180818372252089599, −0.42456785024167406924971147747, 0.42456785024167406924971147747, 1.88039250670180818372252089599, 2.80398561467016136394838585001, 3.33733236362799977187400306883, 4.52453576156045531978634360522, 5.10922599610315259708252606383, 5.81750765150096166009137204014, 6.30144389618192892422436927345, 7.42105783338354734660765287895, 7.903747642917268072119113425258

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