Properties

Label 2-7448-1.1-c1-0-94
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  − 1.34·3-s + 2.90·5-s − 1.20·9-s + 4.99·11-s + 1.55·13-s − 3.90·15-s + 6.19·17-s + 19-s + 6.78·23-s + 3.46·25-s + 5.63·27-s + 0.707·29-s + 9.56·31-s − 6.70·33-s + 4.24·37-s − 2.08·39-s − 2·41-s − 1.25·43-s − 3.49·45-s − 12.3·47-s − 8.30·51-s + 0.891·53-s + 14.5·55-s − 1.34·57-s + 10.5·59-s − 11.8·61-s + 4.51·65-s + ⋯
L(s)  = 1  − 0.774·3-s + 1.30·5-s − 0.400·9-s + 1.50·11-s + 0.430·13-s − 1.00·15-s + 1.50·17-s + 0.229·19-s + 1.41·23-s + 0.692·25-s + 1.08·27-s + 0.131·29-s + 1.71·31-s − 1.16·33-s + 0.697·37-s − 0.333·39-s − 0.312·41-s − 0.191·43-s − 0.520·45-s − 1.80·47-s − 1.16·51-s + 0.122·53-s + 1.95·55-s − 0.177·57-s + 1.37·59-s − 1.51·61-s + 0.559·65-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

\(L(1)\) \(\approx\) \(2.600344035\)
\(L(\frac12)\) \(\approx\) \(2.600344035\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.34T + 3T^{2} \)
5 \( 1 - 2.90T + 5T^{2} \)
11 \( 1 - 4.99T + 11T^{2} \)
13 \( 1 - 1.55T + 13T^{2} \)
17 \( 1 - 6.19T + 17T^{2} \)
23 \( 1 - 6.78T + 23T^{2} \)
29 \( 1 - 0.707T + 29T^{2} \)
31 \( 1 - 9.56T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 1.25T + 43T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 - 0.891T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 1.98T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 7.00T + 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 + 0.494T + 83T^{2} \)
89 \( 1 + 3.38T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−7.933065239126113295323350343336, −6.82690799925581510162544427667, −6.40468210894774872754857294293, −5.87094215674474489217002964211, −5.25243609285043212713477538233, −4.56244067009290530034296602771, −3.42062987259882322499521624492, −2.75882413259309349605950079167, −1.45500311099123761259675854012, −0.979391882788534858588754955441, 0.979391882788534858588754955441, 1.45500311099123761259675854012, 2.75882413259309349605950079167, 3.42062987259882322499521624492, 4.56244067009290530034296602771, 5.25243609285043212713477538233, 5.87094215674474489217002964211, 6.40468210894774872754857294293, 6.82690799925581510162544427667, 7.933065239126113295323350343336

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