Properties

Label 2-448-1.1-c1-0-10
Degree $2$
Conductor $448$
Sign $-1$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s − 3·9-s − 4·11-s − 2·13-s − 6·17-s + 8·19-s − 25-s − 6·29-s − 8·31-s − 2·35-s + 2·37-s + 2·41-s − 4·43-s + 6·45-s + 8·47-s + 49-s − 6·53-s + 8·55-s + 6·61-s − 3·63-s + 4·65-s − 4·67-s + 8·71-s + 10·73-s − 4·77-s − 16·79-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 9-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.894·45-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 0.768·61-s − 0.377·63-s + 0.496·65-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.455·77-s − 1.80·79-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-1$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{448} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−11.05227520706172484192871939077, −9.688287404638284990728895668207, −8.749304120934004278681759852831, −7.79292083571821938290976650763, −7.26382021666693896035142333427, −5.69394971603405508863657314834, −4.92225264716551837532790929786, −3.60173413237973468362473708205, −2.40145156924384381758797773878, 0, 2.40145156924384381758797773878, 3.60173413237973468362473708205, 4.92225264716551837532790929786, 5.69394971603405508863657314834, 7.26382021666693896035142333427, 7.79292083571821938290976650763, 8.749304120934004278681759852831, 9.688287404638284990728895668207, 11.05227520706172484192871939077

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