Properties

Label 2-7448-1.1-c1-0-39
Degree $2$
Conductor $7448$
Sign $1$
Analytic cond. $59.4725$
Root an. cond. $7.71184$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·9-s + 2·11-s − 13-s + 5·17-s − 19-s − 23-s − 5·25-s + 5·27-s − 3·29-s − 4·31-s − 2·33-s + 2·37-s + 39-s + 8·41-s − 8·43-s + 8·47-s − 5·51-s + 9·53-s + 57-s − 59-s − 14·61-s + 13·67-s + 69-s + 10·71-s − 9·73-s + 5·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s + 0.603·11-s − 0.277·13-s + 1.21·17-s − 0.229·19-s − 0.208·23-s − 25-s + 0.962·27-s − 0.557·29-s − 0.718·31-s − 0.348·33-s + 0.328·37-s + 0.160·39-s + 1.24·41-s − 1.21·43-s + 1.16·47-s − 0.700·51-s + 1.23·53-s + 0.132·57-s − 0.130·59-s − 1.79·61-s + 1.58·67-s + 0.120·69-s + 1.18·71-s − 1.05·73-s + 0.577·75-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: yes
Arithmetic: yes
Character: $\chi_{7448} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7448,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.273644394\)
\(L(\frac12)\) \(\approx\) \(1.273644394\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.80486034655764969138116058221, −7.23910078636859994681221968047, −6.35299960280734523899921713070, −5.74288595370304519796460348012, −5.32355758884257144120317337078, −4.30091550407695203443758557978, −3.61659212066604466774296270346, −2.72099923725783562998488470568, −1.70823165715129788843078515348, −0.58570958917079202612575056240, 0.58570958917079202612575056240, 1.70823165715129788843078515348, 2.72099923725783562998488470568, 3.61659212066604466774296270346, 4.30091550407695203443758557978, 5.32355758884257144120317337078, 5.74288595370304519796460348012, 6.35299960280734523899921713070, 7.23910078636859994681221968047, 7.80486034655764969138116058221

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