Properties

Label 2-7448-1.1-c1-0-87
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  + 3.04·3-s − 2.60·5-s + 6.26·9-s + 3.68·11-s + 6.17·13-s − 7.92·15-s − 6.92·17-s − 19-s − 2.61·23-s + 1.77·25-s + 9.94·27-s + 3.12·29-s + 6.97·31-s + 11.2·33-s + 0.292·37-s + 18.7·39-s + 5.29·41-s − 1.96·43-s − 16.3·45-s + 7.99·47-s − 21.0·51-s + 1.79·53-s − 9.58·55-s − 3.04·57-s + 7.12·59-s − 2.14·61-s − 16.0·65-s + ⋯
L(s)  = 1  + 1.75·3-s − 1.16·5-s + 2.08·9-s + 1.11·11-s + 1.71·13-s − 2.04·15-s − 1.67·17-s − 0.229·19-s − 0.546·23-s + 0.354·25-s + 1.91·27-s + 0.579·29-s + 1.25·31-s + 1.95·33-s + 0.0480·37-s + 3.00·39-s + 0.826·41-s − 0.299·43-s − 2.43·45-s + 1.16·47-s − 2.95·51-s + 0.246·53-s − 1.29·55-s − 0.403·57-s + 0.927·59-s − 0.275·61-s − 1.99·65-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

\(L(1)\) \(\approx\) \(3.784615852\)
\(L(\frac12)\) \(\approx\) \(3.784615852\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.04T + 3T^{2} \)
5 \( 1 + 2.60T + 5T^{2} \)
11 \( 1 - 3.68T + 11T^{2} \)
13 \( 1 - 6.17T + 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
23 \( 1 + 2.61T + 23T^{2} \)
29 \( 1 - 3.12T + 29T^{2} \)
31 \( 1 - 6.97T + 31T^{2} \)
37 \( 1 - 0.292T + 37T^{2} \)
41 \( 1 - 5.29T + 41T^{2} \)
43 \( 1 + 1.96T + 43T^{2} \)
47 \( 1 - 7.99T + 47T^{2} \)
53 \( 1 - 1.79T + 53T^{2} \)
59 \( 1 - 7.12T + 59T^{2} \)
61 \( 1 + 2.14T + 61T^{2} \)
67 \( 1 + 7.63T + 67T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 - 8.82T + 73T^{2} \)
79 \( 1 + 6.39T + 79T^{2} \)
83 \( 1 - 7.35T + 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + 8.90T + 97T^{2} \)
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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

−8.187965599414505207824575884336, −7.35514371990263686575467550280, −6.68867316201325482733641443791, −6.10081378781812027440829794007, −4.49200446907709010609490758180, −4.08620130416854187966385880208, −3.67759925714835914760876808775, −2.81679607355224842747371682823, −1.94268461900725684328179818425, −0.937438010895845612405508485826, 0.937438010895845612405508485826, 1.94268461900725684328179818425, 2.81679607355224842747371682823, 3.67759925714835914760876808775, 4.08620130416854187966385880208, 4.49200446907709010609490758180, 6.10081378781812027440829794007, 6.68867316201325482733641443791, 7.35514371990263686575467550280, 8.187965599414505207824575884336

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