Properties

Label 2-7448-1.1-c1-0-28
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.59·3-s + 1.66·5-s + 3.73·9-s + 0.145·11-s − 1.47·13-s − 4.32·15-s − 5.90·17-s + 19-s + 6.22·23-s − 2.22·25-s − 1.90·27-s − 3.13·29-s − 0.882·31-s − 0.377·33-s − 1.96·37-s + 3.83·39-s − 2·41-s − 2.05·43-s + 6.22·45-s − 0.174·47-s + 15.3·51-s + 8.43·53-s + 0.242·55-s − 2.59·57-s − 1.58·59-s − 8.55·61-s − 2.46·65-s + ⋯
L(s)  = 1  − 1.49·3-s + 0.745·5-s + 1.24·9-s + 0.0438·11-s − 0.410·13-s − 1.11·15-s − 1.43·17-s + 0.229·19-s + 1.29·23-s − 0.444·25-s − 0.367·27-s − 0.582·29-s − 0.158·31-s − 0.0656·33-s − 0.322·37-s + 0.614·39-s − 0.312·41-s − 0.314·43-s + 0.927·45-s − 0.0254·47-s + 2.14·51-s + 1.15·53-s + 0.0326·55-s − 0.343·57-s − 0.205·59-s − 1.09·61-s − 0.305·65-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

\(L(1)\) \(\approx\) \(0.9578118909\)
\(L(\frac12)\) \(\approx\) \(0.9578118909\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.59T + 3T^{2} \)
5 \( 1 - 1.66T + 5T^{2} \)
11 \( 1 - 0.145T + 11T^{2} \)
13 \( 1 + 1.47T + 13T^{2} \)
17 \( 1 + 5.90T + 17T^{2} \)
23 \( 1 - 6.22T + 23T^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 + 0.882T + 31T^{2} \)
37 \( 1 + 1.96T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 2.05T + 43T^{2} \)
47 \( 1 + 0.174T + 47T^{2} \)
53 \( 1 - 8.43T + 53T^{2} \)
59 \( 1 + 1.58T + 59T^{2} \)
61 \( 1 + 8.55T + 61T^{2} \)
67 \( 1 - 1.33T + 67T^{2} \)
71 \( 1 - 7.37T + 71T^{2} \)
73 \( 1 - 4.72T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 4.37T + 83T^{2} \)
89 \( 1 + 1.39T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−7.65921178167109177008232676100, −6.90878494550371813501029955365, −6.46919398502530767013496533129, −5.78813022301574740028019494471, −5.12702259075710936439266292829, −4.69693376172006113942445566761, −3.69312078590170298680356070125, −2.50910114169373286090819234576, −1.65922171927053373976948418559, −0.53325613418273381647312438425, 0.53325613418273381647312438425, 1.65922171927053373976948418559, 2.50910114169373286090819234576, 3.69312078590170298680356070125, 4.69693376172006113942445566761, 5.12702259075710936439266292829, 5.78813022301574740028019494471, 6.46919398502530767013496533129, 6.90878494550371813501029955365, 7.65921178167109177008232676100

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