Properties

Label 2-448-1.1-c5-0-46
Degree $2$
Conductor $448$
Sign $-1$
Analytic cond. $71.8519$
Root an. cond. $8.47655$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.95·3-s − 11.6·5-s − 49·7-s − 207.·9-s + 12.4·11-s + 8.71·13-s − 69.5·15-s + 2.12e3·17-s + 2.22e3·19-s − 291.·21-s + 2.94e3·23-s − 2.98e3·25-s − 2.68e3·27-s − 2.46e3·29-s − 9.13e3·31-s + 73.8·33-s + 572.·35-s − 6.68e3·37-s + 51.8·39-s + 1.13e4·41-s − 1.18e4·43-s + 2.42e3·45-s + 4.12e3·47-s + 2.40e3·49-s + 1.26e4·51-s − 3.68e4·53-s − 145.·55-s + ⋯
L(s)  = 1  + 0.381·3-s − 0.208·5-s − 0.377·7-s − 0.854·9-s + 0.0309·11-s + 0.0143·13-s − 0.0798·15-s + 1.78·17-s + 1.41·19-s − 0.144·21-s + 1.16·23-s − 0.956·25-s − 0.708·27-s − 0.544·29-s − 1.70·31-s + 0.0118·33-s + 0.0789·35-s − 0.802·37-s + 0.00546·39-s + 1.05·41-s − 0.974·43-s + 0.178·45-s + 0.272·47-s + 0.142·49-s + 0.681·51-s − 1.80·53-s − 0.00646·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-1$
Analytic conductor: \(71.8519\)
Root analytic conductor: \(8.47655\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 448,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 49T \)
good3 \( 1 - 5.95T + 243T^{2} \)
5 \( 1 + 11.6T + 3.12e3T^{2} \)
11 \( 1 - 12.4T + 1.61e5T^{2} \)
13 \( 1 - 8.71T + 3.71e5T^{2} \)
17 \( 1 - 2.12e3T + 1.41e6T^{2} \)
19 \( 1 - 2.22e3T + 2.47e6T^{2} \)
23 \( 1 - 2.94e3T + 6.43e6T^{2} \)
29 \( 1 + 2.46e3T + 2.05e7T^{2} \)
31 \( 1 + 9.13e3T + 2.86e7T^{2} \)
37 \( 1 + 6.68e3T + 6.93e7T^{2} \)
41 \( 1 - 1.13e4T + 1.15e8T^{2} \)
43 \( 1 + 1.18e4T + 1.47e8T^{2} \)
47 \( 1 - 4.12e3T + 2.29e8T^{2} \)
53 \( 1 + 3.68e4T + 4.18e8T^{2} \)
59 \( 1 + 1.64e3T + 7.14e8T^{2} \)
61 \( 1 + 4.19e4T + 8.44e8T^{2} \)
67 \( 1 + 5.07e4T + 1.35e9T^{2} \)
71 \( 1 + 2.33e4T + 1.80e9T^{2} \)
73 \( 1 - 5.29e4T + 2.07e9T^{2} \)
79 \( 1 + 6.24e4T + 3.07e9T^{2} \)
83 \( 1 - 7.74e3T + 3.93e9T^{2} \)
89 \( 1 - 5.08e3T + 5.58e9T^{2} \)
97 \( 1 - 9.86e4T + 8.58e9T^{2} \)
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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

−9.618957067460938228257110015284, −9.099126098221830788165577251409, −7.87881801152087704130131554303, −7.33984799644811555007829863165, −5.90059933317522316969077196009, −5.21654159103982495045597116349, −3.56722906778692444289648931908, −3.02658373179114218231606444482, −1.43279532819832393794604798986, 0, 1.43279532819832393794604798986, 3.02658373179114218231606444482, 3.56722906778692444289648931908, 5.21654159103982495045597116349, 5.90059933317522316969077196009, 7.33984799644811555007829863165, 7.87881801152087704130131554303, 9.099126098221830788165577251409, 9.618957067460938228257110015284

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